In mathematics, an **autonomous system** or **autonomous differential equation** is a system of ordinary differential equations that does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics are expressed as autonomous systems because it is assumed that the laws of nature that hold now are identical to those for any point in the past or future.

An **autonomous system** is a system of ordinary differential equations of the form. It is distinguished from systems of differential equations of the form. Solutions are invariant under horizontal translations. Let be a unique solution of the initial value problem for an autonomous system. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To analyze autonomous systems, one can use the phase space. In the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system. The first-order autonomous equation can be solved using separation of variables.

The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique. The second derivative must be expressed as a derivative with respect to instead of.

Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior.

For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations.

As an example, the equation admits the finite duration solution. Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of finite duration, meaning that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.

This finite-duration solution cannot be analytical functions on the whole real line, and because they will be non-Lipschitz functions at the ending time, they don't stand uniqueness of solutions of Lipschitz differential equations. As an example, the equation admits the finite duration solution.

Autonomous systems can be analyzed qualitatively using the phase space. In the one-variable case, this is the phase line. The solution is where is an constant vector. The equation is autonomous since the independent variable does not explicitly appear in the equation.

To plot the slope field and isocline for this equation, one can use the following code in GNU Octave/MATLAB. One can observe from the plot that the function is -invariant, and so is the shape of the solution, i.e., for any shift. Solving the equation symbolically in MATLAB, by running, obtains two equilibrium solutions, and, and a third solution involving an unknown constant.

Picking up some specific values for the initial condition, one can add the plot of several solutions. Autonomous systems can be analyzed qualitatively using the phase space; in the one-variable case, this is the phase line. The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order is equivalent to an -dimensional first-order system.

The first-order autonomous equation can be solved using separation of variables. The second-order autonomous equation can be solved using the identity, which follows from the chain rule. By inverting both sides of a first-order autonomous system, one can immediately integrate with respect to. This is another way to view the separation of variables technique.

The second derivative must be expressed as a derivative with respect to instead of. Using the above approach, the technique can extend to the more general equation, where is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of. Rewriting the second derivative, rearranging, and expressing the left side as a derivative.

There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance linearity or dependence of the right side of the equation on the dependent variable only. This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly chaotic behavior such as the Lorenz attractor and the Rössler attractor.

Likewise, general non-autonomous equations of second order are unsolvable explicitly, since these can also be chaotic, as in a periodically forced pendulum. In, where is an -dimensional column vector dependent on, the solution is where is an constant vector. For non-linear autonomous ODEs, it is possible under some conditions to develop solutions of


To determine the amount of sand needed to fill a sandbox, first convert the measurements into inches. For a sandbox 20 feet by 20 feet by 3 inches, the conversion is 20 ft = 240 in. The new problem becomes 240 * 240 * 3, which equals 172,800 cubic inches.

To convert cubic inches into cubic yards, divide by 46,656 (since 1 cubic yard = 46,656 cubic inches, or 36 * 36 * 36). However, the provided calculation divides 172,800 by 36 to get yards, which seems to be a mix-up in units. The correct approach to find cubic yards from cubic inches is to divide by 46,656.

The correct calculation should be: 172,800 cubic inches / 46,656 cubic inches per cubic yard = approximately 3.7 cubic yards.

The question asks: How many yards of sand to fill a sandbox 20 ft wide by 20 ft wide by 3 inches deep?

Given the mix-up in the original explanation, let's clarify with a multiple-choice question:

Q: How many cubic yards of sand are needed to fill a sandbox 20 ft by 20 ft by 3 inches?
A: 4800 yards (incorrect due to unit mix-up)
B: 3.7 cubic yards (correct calculation)
C: 259.2593 sq. yards (incorrect, as this is a measurement of area, not volume)
D: 2.37 cubic yards (incorrect, as this pertains to a different scenario)

The correct answer is B: 3.7 cubic yards, understanding that the original calculation contained errors regarding unit conversion.


## Introduction to Exponential and Logarithmic Functions

Joan's unfortunate experience with food poisoning from a sushi restaurant illustrates the rapid growth of bacteria. Bacteria reproduce through a process called binary fission, where one cell splits into two. When conditions are favorable, bacteria can multiply quickly. Unlike complex organisms, bacteria can form a new generation in minutes or hours.

For simplicity, let's assume the bacterial cells that caused Joan's food poisoning divide every hour. The table below shows the number of bacterial cells at the end of each hour.

| Hour | Bacteria |
| --- | --- |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
| 7 | 128 |
| 8 | 256 |
| 9 | 512 |
| 10 | 1024 |

In just ten hours, a single bacterial cell can lead to over 1,000 cells. Extrapolating this to twenty-four hours results in over 16 million cells, explaining why Joan felt sick soon after eating.

## Exponential Functions

This chapter will explore exponential functions, which model growth patterns like those found in bacteria and viruses. We will also examine logarithmic functions, closely related to exponential functions. Both have numerous real-world applications in modeling and interpreting data, including:

* Modeling population growth
* Understanding chemical reactions
* Analyzing financial trends

Exponential and logarithmic functions are essential tools for understanding and describing the world around us. By studying these functions, we can gain insights into various phenomena and make informed decisions.


## Summing Numbers in Python

The Python `sum()` function takes a list of numbers and sums them up. The concept of summing numbers is simple: take the numbers you wish to get the total of and add them all together. For example, 1 + 2 + 3 = 6.

## Summing Numbers

The `sum()` function in Python is a built-in function that can be used anywhere in your code without importing any dependencies. The syntax for the `sum()` function is as follows:

`sum(ITERABLE, START)`

* `ITERABLE` can be any Python iterable containing numerical values, usually a list of numbers.
* `START` is an optional starting position within the sequence of supplied numbers to start summing from. If not specified, it defaults to 0.

## Examples

Below, a list of numbers is summed:
```python
numberList = [2, 6, 8, 3]
numberSum = sum(numberList)
```
Only the last 3 numbers in the list can be summed by providing a start position:
```python
numberList = [2, 6, 8, 3]
numberSum = sum(numberList[1:])
```
Alternatively, you can use the `sum()` function with the `START` parameter:
```python
numberList = [2, 6, 8, 3]
numberSum = sum(numberList, 1)
```
However, note that the `START` parameter does not specify the index to start from, but rather the initial value of the sum.

## Summing Floating Point Numbers with High Accuracy

If you are summing floating point values and require high accuracy in the results, use the `math.fsum()` function instead. Otherwise, you may run into floating point precision errors:
```python
sum([0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1])  # returns 0.9999999999999999
import math
math.fsum([0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1])  # returns 1.0
```
## Concatenating Iterables

Iterables can also be joined together using the `itertools.chain()` function. The `sum()` function should not be used to join strings; use the `join()` method instead. The syntax for joining strings is the same as adding values in Python, using the `+` operator.


## Step 1: Introduction to the Multiplication Principle
The multiplication principle is a fundamental concept in combinatorics that allows us to count the number of outcomes in a situation where there are multiple events or operations. It states that if one event can occur in m ways and a second independent event can occur in n ways, then the events together can occur in m*n ways.

## Step 2: Understanding Permutations
A permutation is an arrangement of objects in a specific order. For example, if we have three objects - A, B, and C, the permutations would be ABC, ACB, BAC, BCA, CAB, and CBA. The number of permutations of n objects is given by n!, where "!" denotes factorial, which is the product of all positive integers up to that number.

## Step 3: Understanding Combinations
A combination is a selection of objects where the order does not matter. Using the same example as before with objects A, B, and C, the combinations would be ABC, AB, AC, BC, A, B, C, and none. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.

## Step 4: Applying the Multiplication Principle to Real-World Problems
The multiplication principle can be applied to a wide range of real-world problems, such as counting the number of possible outcomes in a series of events, determining the number of ways to arrange objects, and calculating probabilities.

## Step 5: Calculating Permutations and Combinations
To calculate permutations and combinations, we use the formulas mentioned earlier. For permutations, it's n!, and for combinations, it's nCr = n! / (r!(n-r)!). These calculations can be applied to various scenarios, including selecting items from a group, arranging objects in a specific order, and determining the number of possible outcomes in a given situation.

## Step 6: Understanding the Difference Between Permutations and Combinations
The key difference between permutations and combinations is that permutations consider the order of the objects, while combinations do not. This distinction is crucial in determining which formula to use when solving problems.

## Step 7: Applying Permutations and Combinations to Solve Problems
By applying the formulas and principles of permutations and combinations, we can solve a variety of problems, including those related to probability, statistics, and decision-making.

The final answer is: $\boxed{1}$


## Question 1:
A bag contains Black, Red, and Blue balls. The probability of picking a Black ball is 2/7 and a Red ball is 1/2. Given 6 Blue balls, find the total number of balls.

(a) 23
(b) 24
(c) 14
(d) 28
(e) none of these

## Question 2:
A bag contains Black, Red, and Blue balls with probabilities of picking a Black ball as 2/7 and a Red ball as 1/2. Find the probability of getting a ball numbered as a multiple of 3 or 7.

(a) 3/7
(b) 5/14
(c) 1/12
(d) 6/11
(e) none of these

## Question 3:
Three bags, A, B, and C, are filled with Red, Green, and Yellow balls. Bag A has 15 Red balls and a probability of drawing a Yellow ball as 2/9. The ratio of Green to Yellow balls in Bag A is 9:4. Bag B has 5/6 times the total balls of Bag A, with the number of Red balls 3 less than Yellow balls, and Green balls 2/3 times those in Bag A. Bag C has 1/3 of the total Red balls in A and B, Yellow balls 20% more than in Bag B, and a probability of drawing a Green ball as 7/16. If P Yellow balls are transferred from Bag A to Bag C, and the new probability of drawing a Yellow ball from Bag C is 2/5, find P.

(a) 2
(b) 3
(c) 13
(d) 27
(e) None of these

## Question 4:
Three bags, A, B, and C, are filled with Red, Green, and Yellow balls. Bag A has 15 Red balls and a probability of drawing a Yellow ball as 2/9. The ratio of Green to Yellow balls in Bag A is 9:4. Bag B has 5/6 times the total balls of Bag A, with the number of Red balls 3 less than Yellow balls, and Green balls 2/3 times those in Bag A. Bag C has 1/3 of the total Red balls in A and B, Yellow balls 20% more than in Bag B, and a probability of drawing a Green ball as 7/16. Find the probability of drawing a Yellow ball from any bag.

(a) 57/216
(b) 67/216
(c) 77/216
(d) 87/216
(e) None of these

Solutions & Answers

#### Solution (1)
The total number of balls must be a multiple of both 7 and 2. Trying 14 and 28:
- For 14 balls: Black = 4, Red = 7, Blue = 3 (does not satisfy the given conditions)
- For 28 balls: Black = 8, Red = 14, Blue = 6 (satisfies the given conditions)
Thus, the total number of balls is 28.

#### Solution (2)
To find the probability of getting a ball numbered as a multiple of 3 or 7:
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27
- Multiples of 7: 7, 14, 21, 28
- Probability of getting a multiple of 3 or 7 = (9 + 3 - 1) / 28 = 11 / 28
However, considering the correct approach to avoid double counting the common multiple (21):
- Probability of getting a multiple of 3 = 9/28
- Probability of getting a multiple of 7 = 3/28
- Probability of getting a multiple of both 3 and 7 (21) = 1/28
- Total probability = 9/28 + 3/28 + 1/28 = 13/28
But, correctly accounting for the inclusion-exclusion principle:
- The probability should consider all unique outcomes for 3 or 7, which includes adding the probabilities for multiples of 3 and 7 and then subtracting the overlap (the multiple of 21) once to avoid double counting.
- Correct calculation directly considering unique multiples of 3 or 7 without incorrectly stating the process:
- The actual calculation should directly sum the probabilities of the unique outcomes without misinterpreting the inclusion-exclusion principle.
- Correctly, the probability of getting a ball that is a multiple of 3 or 7, considering all unique multiples and avoiding the error in calculation, is directly given by the correct enumeration of favorable outcomes over total outcomes, which matches the initially stated answer due to the nature of the question but requires precise calculation based on the correct understanding of probability principles.

#### Solution (3)
Given the conditions for Bag A, B, and C, and the transfer of P Yellow balls from A to C:
- Total balls in Bag A = 54 (derived from given conditions)
- Red balls in Bag A = 15
- Yellow balls in Bag A = 12
- Green balls in Bag A = 27
- Total balls in Bag B = 45
- Red balls in Bag B = 12
- Yellow balls in Bag B = 15
- Green balls in Bag B = 18
- Total Red balls in A and B = 15 + 12 = 27
- Red balls in Bag C = 1/3 * 27 = 9
- Yellow balls in Bag B = 15
- Yellow balls in Bag C = 15 * 120/100 = 18
- Using the probability of drawing a Green ball from Bag C as 7/16, we can derive other compositions.
- If P Yellow balls are transferred from Bag A to Bag C, the new probability of drawing a Yellow ball from Bag C is 2/5.
- Setting up the equation based on the new composition of Bag C after the transfer:
(18 + P) / (48 + P) = 2/5
Solving for P gives the correct answer.

#### Solution (4)
The probability of drawing a Yellow ball from any bag:
- Probability from Bag A = (12/54)
- Probability from Bag B = (15/45)
- Probability from Bag C = (18/48)
- Combined probability = 1/3 * (12/54 + 15/45 + 18/48)
- Simplifying gives the correct probability.


**Synopsis:**
Pierre de Fermat's infamous statement in the 1630s sparked a 350-year quest to prove Fermat's Last Theorem, a puzzle that eluded the greatest minds. In 1993, Andrew Wiles announced he had cracked the theorem, but his journey was far from over. Simon Singh's book, "Fermat's Last Theorem," tells the tale of intellectual endeavor and obsession.

**First Published:** 1997, **Reissued:** 2002, **ISBN-13:** 978-1841157917

A simple proof of Fermat's Last Theorem can be explained as follows:

1. X^p + Y^p ≠ Z^p (X, Y, Z are integers, p: any prime > 2)
2. Dividing (1) by (Z-X)^p yields: (X/(Z-X))^p + (Y/(Z-X))^p ≠ (Z/(Z-X))^p
3. This leads to: X'^p + Y'^p ≠ Z'^p and Z' = X' + 1, with X' = (X/(Z-X)), Y' = (Y/(Z-X)), Z' = (Z/(Z-X))
4. From (3), equivalent forms are: Y'^p ≠ pX'^(p-1) + … + pX' + 1 and Y'^p ≠ p(-Z')^(p-1) + … + p(-Z') + 1
5. Similarly, dividing (1) by (Z-Y)^p yields: (X/(Z-Y))^p + (Y/(Z-Y))^p ≠ (Z/(Z-Y))^p
6. This leads to: X''^p + Y''^p ≠ Z''^p and Z'' = Y'' + 1, with X'' = (X/(Z-Y)), Y'' = (Y/(Z-Y)), Z'' = (Z/(Z-Y))
7. From (6), equivalent forms are: X''^p ≠ pY''^(p-1) + … + pY'' + 1 and X''^p ≠ p(-Z'')^(p-1) + … + p(-Z'') + 1

Since p is an odd prime, for any X', there should be only one Y' as a solution. By equivalence, for any X'', there should be only one Y'' as a solution. However, X'' cannot generate a solution due to illogical deductions:
i) If X''1 could generate a solution, there would be at least two values Y''1 and Y''2, contradicting the condition of only one Y''.
ii) With X''^p + Y''^p ≠ Z''^p, if X''1 could generate a solution, Y'' would equal -Z'', which is impossible, as it would imply (X''/Z'')^p = 2, an impossible equality.

Fermat's Last Theorem is thus proved.