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Example 1. Given the differential equation $y^{\prime}=x$. Construct the direction field and indicate the approximate general solution of this equation. | Solution. We have $f(x, y)=x$. This function is defined for all values of $x$ and $y$ (it does not depend on $y$). This means that $D=\mathbb{R}^{2}$, i.e., the domain $D$ is the entire plane $O x y$. At all points on the plane where $x=0$ (this is the $O y$ axis), we have $y^{\prime}=0$. This means that at points on t... | \frac{x^{2}}{2}+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,305 |
Example 2. Given the differential equation
$$
\left(x-x^{3}\right) y^{\prime}+\left(2 x^{2}-1\right) y-x^{3}=0
$$
Check whether the function $y=x \sqrt{1-x^{2}}+x$ is a solution to this equation. Will it be its general solution? | Solution. We find $y^{\prime}=\sqrt{1-x^{2}}-\frac{x^{2}}{\sqrt{1-x^{2}}}+1$.
Expressions for $y$ and $y^{\prime}$ will be substituted into the left side of the original equation. After transformations, we get:
$$
\begin{aligned}
\left(x-x^{3}\right) & \left(\sqrt{1-x^{2}}-\frac{x^{2}}{\sqrt{1-x^{2}}}+1\right)+\left(... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,306 |
Example 3. Check that the function given parametrically:
$$
\left\{\begin{array}{l}
x=\ln t+\sin t \\
y=t(1+\sin t)+\cos t
\end{array}\right.
$$
is a solution to the differential equation $x=\ln y^{\prime}+\sin y^{\prime}$. | Solution. We find $y^{\prime}$ by the rule of differentiating a parametrically defined function:
$$
y^{\prime}=y_{x}^{\prime}=\frac{y^{\prime}(t)}{x^{\prime}(t)}
$$
Differentiating:
$$
\begin{aligned}
& x^{\prime}(t)=\frac{1}{t}+\cos t, \quad y^{\prime}(t)=1+\sin t+t \cos t-\sin t=t \cos t+1 \\
& y^{\prime}=y_{x}^{\... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,307 |
Example 4. Check that the function $y=x\left(e^{x^{2}}+C\right)$, depending on an arbitrary constant $C$, is the general solution of the differential equation $y^{\prime}=\frac{y}{x}+2 x^{2} \cdot e^{x^{2}}$. | Solution. We have $y=x e^{x^{2}}+C x, y^{\prime}=e^{x^{2}}+2 x^{2} \cdot e^{x^{2}}+C$.
After substitution into the equation, we obtain an identity (verify independently) for all $x \neq 0$. For any values of $C$, the given function is a solution to the original equation.
Note. Let $x_{0} \neq 0$ and $y_{0}$ be arbitr... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,308 |
Example 5. Construct (approximately) the direction field and integral curves for the differential equation $y^{\prime}=x^{2}+y^{2}$. | Solution. The right-hand side of the equation is defined throughout the plane $O x y$, i.e., the domain $D$ mentioned in the existence and uniqueness theorem is the entire plane $O x y$. At the origin, we have $x=y=0$ and $y^{\prime}=0$. In no other points is $y^{\prime}=0$, i.e., the origin is the only point with a ho... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,309 |
Example 6. Construct a differential equation if its general solution is known to be $y=\frac{1}{6} x^{4}+\frac{C}{x^{2}}$. | Solution. We will use the definition: we need to form an equation containing the argument $x$, the unknown function $y$, and its derivative $y'$. For this, we will differentiate the given function, and then from the two equations combined into a system, we will eliminate the constant $C$. So,
$$
\left\{\begin{array}{l... | y'+2\frac{y}{x}=x^{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,310 |
Example 1. Solve the differential equation $2 x d x-\frac{d y}{\cos ^{2} y}=$ $=0$. Also find the particular solution satisfying the initial condition $y=0$ when $x=0$. | Solution. We have the equation with separated variables: $\frac{d y}{\cos ^{2} y}=2 x d x$. We integrate each part by its own variable: $\int \frac{d y}{\cos ^{2} y}=\int 2 x d x$, i.e., $\operatorname{tg} y=x^{2}+C$. We have obtained the general integral, or the general solution in implicit form. From this, we get the... | \operatorname{arctg}x^{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,311 |
Example 2. Solve the equation $x e^{y} d x+\left(y+y x^{2}\right) d y=0$. | Solution. By factoring out, we transform the equation into a separable equation: $x e^{y} d x+y\left(1+x^{2}\right) d y=0$. We divide term by term by $\left(1+x^{2}\right) e^{y}$. We obtain the equation with separated variables: $\frac{x d x}{1+x^{2}}+y e^{-y} d y=0$. We integrate (the second term by parts): $\int \fra... | \ln\sqrt{1+x^{2}}-(y+1)e^{-y}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,312 |
Example 3. Solve the equation $y^{\prime}=\frac{2 x+3 y-5}{4 x+6 y+8}$. | Solution. Obviously, the right-hand side of the given equation is a function of $2 x+3 y$. Let $v=2 x+3 y$. This equality is differentiable with respect to the variable $x$ ( $x$ is an argument for both the function $y$ and the function $v$ ): $2+3 y^{\prime}=v^{\prime}$.
From this, we find $y^{\prime}=\frac{1}{3}\lef... | 3y+\frac{27}{7}\ln|7(2x+3y)+1|=\frac{3}{2}x+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,313 |
Example 4. Solve the equation $y^{\prime}-y^{2}=4 x^{2}+4 x y+2$. | Solution. Rewriting the given equation in the form $y^{\prime}=(2 x+y)^{2}+2$, we notice the appropriateness of the substitution (change of variable) $2 x+y=v$. Differentiating this equality (with respect to the variable $x$), we get $2+y^{\prime}=v^{\prime}$, or $y^{\prime}=v^{\prime}-2$. The original equation takes t... | 2\operatorname{tg}(2x+C)-2x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,314 |
Example 5. Integrate the differential equation $y^{\prime}=\frac{x \sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}\left(y+e^{y}\right)}$ and also find the particular solution that satisfies the Cauchy condition: $x_{0}=2, y_{0}=1$. | Solution. We transform the given equation to separate the variables (replacing $y^{\prime}=\frac{d y}{d x}$):
$$
\frac{d y}{d x}=\frac{x \sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}\left(y+e^{y}\right)}, \quad\left(y+e^{y}\right) d y=\frac{x \sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}} \cdot d x
$$
We integrate the obtained equality: $\int... | \frac{y^{2}-x^{2}}{2}+e^{y}-\ln\frac{x+\sqrt{1+x^{2}}}{2+\sqrt{5}}=e-\frac{3}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,315 |
Example 6. Solve the differential equation
$$
\left(1+x^{2} y^{2}\right) y+(x y-1)^{2} \cdot x \cdot y^{\prime}=0
$$ | Solution. Obviously, the variables cannot be separated. Let's perform the substitution $x y = z$, i.e., $y = \frac{z}{x}$, where $z$ is a new unknown function of the argument $x$. Differentiating the first equality, we get $y + x y' = z'$. Hence, $x y' = z' - y$. Substituting into the original equation gives:
\[
\left... | xy-\frac{1}{xy}-2\ln|y|=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,316 |
Example 1. Find the general integral of the equation $y^{\prime}=\frac{x^{2}+3 x y-y^{2}}{3 x^{2}-2 x y}$. | Solution. Formally, it is not obvious that the right-hand side of this equation is a function depending on the ratio $\frac{y}{x}$. To see this, factor out $x^2$ in the numerator and the denominator and cancel this factor. We get
$$
y'=\frac{x^{2}\left(1+3 \frac{y}{x}-\left(\frac{y}{x}\right)^{2}\right)}{x^{2}\left(3-... | 3\operatorname{arctg}\frac{y}{x}=\ln\frac{C(x^{2}+y^{2})}{|x|} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,317 |
Example 2. Solve the equation $y^{\prime}=\frac{x+y-1}{2 x-y+4}$. | Solution. This equation would be homogeneous if there were no free terms in the numerator and denominator of the right-hand side. It has the form indicated in point $2^{\circ}$, and can be reduced to a homogeneous equation after additional searching.
From the linear system $\left\{\begin{array}{r}x+y-1=0, \\ 2 x-y+4=0... | \ln((y-2)^{2}-(y-2)(x+1)+(x+1)^{2})-2\sqrt{3}\operatorname{arctg}\frac{2y-x-5}{\sqrt{3}(x+1)}=C_{\mathrm{I}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,318 |
Example 3. Solve the equation $y^{\prime}=\frac{2 x+2 y+5}{x+y-1}$. | Solution. The system $\left\{\begin{array}{l}2 x+2 y+5=0 \\ x+y-1=0\end{array}\right.$ is unsolvable, so the equation cannot be reduced to a homogeneous one. Notice that the right-hand side is a function of $x+y$. The equation can be reduced to one with separable variables (see §2). Let $x+y=z$. Then $2 x+2 y+5=2 z+5$.... | y-2x-\frac{7}{3}\ln|3x+3y+4|=3C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,319 |
Example 1. Solve the equation $y^{\prime}+x y=y^{3} e^{x^{2}}$. | Solution. Let $y=u \cdot v, y^{\prime}=u^{\prime} v+u v^{\prime}$. We get $u^{\prime} v + u v^{\prime} + x u \cdot v = u^{3} \cdot v^{3} \cdot e^{x^{2}}$.
1) From the equation $u v^{\prime} + x u v = 0$, i.e., $\frac{d v}{d x} = -x v$, or $\frac{d v}{v} = -x d x$, we find $\ln |v| = -\frac{x^{2}}{2}$, or $v = e^{-x^{2... | \frac{e^{-x^{2}/2}}{\sqrt{2(C-x)}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,320 |
Example 2. Solve the equation $y^{\prime}=\frac{1}{x \cos y+\sin 2 y}$. | Solution. The given equation is not linear (or a Bernoulli equation), so a substitution of the form $y=u v$ is not applicable. It is necessary to recall the properties of inverse functions and the relationship between their derivatives: $x^{\prime} \cdot y^{\prime}=1, \quad y^{\prime}=\frac{1}{x^{\prime}}$, to replace ... | Ce^{\siny}-2(1+\siny) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,321 |
Example 3. Integrate the differential equation $y^{\prime}=\frac{y}{2 y \ln y+y-x}$. | Solution. We swap the roles of the function $y$ and the argument $x$:
$$
x^{\prime}=\frac{2 y \ln y+y-x}{y}, \quad x^{\prime}+\frac{x}{y}=2 \ln y+1
$$
We obtain a linear equation with respect to the unknown function $x=x(y)$. Assuming $x=u \cdot v, x^{\prime}=u^{\prime} v+u v^{\prime}$, we get the system
$$
\left\{\... | y\lny+\frac{C}{y} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,322 |
Example 1. Solve the equation $(x+y) d x+(x+2 y) d y=0$. | Solution. Here $P=x+y, \quad Q=x+2 y, \quad \frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}=1$. We have an equation in total differentials.
Find the function $u=u(x, y)$ from the condition $d u=(x+y) d x+(x+$ $+2 y) d y$. Since $d u=\frac{\partial u}{\partial x} d x+\frac{\partial u}{\partial y} d y$, then... | \frac{x^{2}}{2}+xy+y^{2}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,323 |
Example 2. Solve the equation $\left(x+y^{2}\right) d x-2 x y d y=0$. | Solution. We have $P=x+y^{2}, Q=-2 x y, \frac{\partial P}{\partial y}=2 y, \frac{\partial Q}{\partial x}=-2 y$.
The equation is not an exact differential equation. The expression $\left(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}\right) \frac{1}{Q}=-\frac{2}{x}$ depends only on the variable $x$, so acc... | \ln|x|-\frac{y^{2}}{x}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,324 |
Example 3. Solve the equation $2 x y \ln y d x+\left(x^{2}+y^{2} \sqrt{y^{2}+1}\right) d y=0$. | Solution. We have $P=2 x y \ln y, \quad Q=x^{2}+y^{2} \sqrt{y^{2}+1}, \quad \frac{\partial P}{\partial y}=$ $=2 x \ln y+2 x, \frac{\partial Q}{\partial x}=2 x, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=-2 x \ln y, \frac{1}{P}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)=-\fr... | 3x^{2}\lny+\sqrt{(y^{2}+1)^{3}}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,325 |
Example 1. $\left(y^{\prime}\right)^{2}+\left(\frac{x-1}{y}+\frac{1+2 x}{\sqrt{1+x+x^{2}}}\right) y^{\prime}+\frac{(x-1)(2 x+1)}{\sqrt{1+x+x^{2}} \cdot y}=0$. | Solution. Analysis of the left side of the equation allows us to factorize it:
$$
\left(y^{\prime}+\frac{2 x+1}{\sqrt{1+x+x^{2}}}\right)\left(y^{\prime}+\frac{x-1}{y}\right)=0
$$
This equation is equivalent to a system of two equations with separable variables:
$$
\left[\begin{array} { l }
{ y ^ { \prime } = - \fra... | C_{1}\cdote^{x}-x-1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,326 |
Example 4. $y=3\left(y^{\prime}\right)^{2}-5\left(y^{\prime}\right)^{3}$.
| Solution. The given equation can be classified as a general form equation $\varphi\left(y, y^{\prime}\right)=0$, and we can apply the reasoning from section 6.2.
Let $y^{\prime}=t\left(t-\right.$ be a parameter). We obtain the equality $y=3 t^{2}-5 t^{3}$, which we differentiate with respect to $x$ (note that $t=y^{\p... | {\begin{pmatrix}-\frac{3}{10}(2-5)^{2}+C_{\mathrm{I}},\\3^{2}-5^{3}\end{pmatrix}\quad0} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,327 |
Example 5. $x\left(1+\left(y^{\prime}\right)^{2}\right)=1$.
| Solution. In fact, this equation can be solved for the derivative $y^{\prime}: y^{\prime}= \pm \sqrt{\frac{1}{x}-1}$, and from this, we can obtain the general solution of the original equation. However, our goal is to illustrate the reasoning of point 6.3. Let $y^{\prime}=t, \quad \frac{d y}{d x}=t$, i.e., $d x=\frac{d... | {\begin{pmatrix}\frac{1}{1+^{2}}\\-\operatorname{arctg}+\frac{}{^{2}+1}+C\end{pmatrix}.} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,328 |
Example 6. Solve the Clairaut equation $y^{\prime}=x y^{\prime}+\frac{3}{y^{\prime 2}}$. | Solution. The Clairaut equation $y=x y^{\prime}+\psi\left(y^{\prime}\right)$ (here $\psi\left(y^{\prime}\right)=\frac{3}{y^{\prime 2}}$) is solved in two stages.
1) Notice that the substitution $y^{\prime}=C(C \neq 0-$ an arbitrary constant) satisfies the given equation: from it follows that $y=x \cdot C+C_{1} ;$ sett... | y_{\text{gen}}=Cx+\frac{3}{C^{2}},\quady_{\text{spec}}=\sqrt[3]{\frac{81}{4}x^{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,329 |
Example 7. Integrate the Lagrange differential equation $y=\frac{3}{2} x y^{\prime}+e^{y^{\prime}}$. | Solution. Let's replace $y^{\prime}=t$. The equation takes the form $y=\frac{3}{2} x t+e^{t}$. This equality is differentiable with respect to $x$, then instead of $y^{\prime}$ we substitute $t$: $y^{\prime}=\frac{3}{2} t+\frac{3}{2} x t^{\prime}+e^{t} \cdot t^{\prime} ; t=\frac{3}{2} t+\frac{3}{2} x t^{\prime}+e^{t} \... | {\begin{pmatrix}\frac{2}{^{3}}(C-(^{2}-2+2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,330 |
Example 1. Solve the equation $y^{IV}=\cos x$. | Solution. The equation belongs to type 1). Integrating:
$$
y^{\prime \prime \prime}=\int \cos x d x, \text { i.e. } y^{\prime \prime \prime}=\sin x+C_{1}
$$
Integrate again: $y^{\prime \prime}=-\cos x+C_{1} x+C_{2}$. Next, by analogy:
$$
y^{\prime}=-\sin x+C_{1} \frac{x^{2}}{2}+C_{2} x+C_{3}
$$
The last integration... | \cosx+C_{1}x^{3}+C_{2}x^{2}+C_{3}x+C_{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,331 |
Example 2. Find the general solution of the equation $\left(1+x^{2}\right) y^{\prime \prime}+y^{\prime 2}+$ $+1=0$ and the particular solution that satisfies the initial conditions $x=0, y=1, y^{\prime}=1$. | Solution. The given equation does not explicitly contain the variable $y$, so according to 2), we will use the substitution $y^{\prime}=p, y^{\prime \prime}=p^{\prime}$. This leads to a separable equation in terms of $p$: $\left(1+x^{2}\right) p^{\prime}+p^{2}+1=0, p^{\prime}+\frac{1+p^{2}}{1+x^{2}}=0$, i.e., $\frac{d ... | -x+2\ln|x+1|+1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,332 |
Example 3. Find the particular solution of the equation $y^{\prime \prime}=y^{\prime} \cdot \cos y$, satisfying the initial conditions $x=1, y=\frac{\pi}{2}, y^{\prime}=1$. | Solution. Visually, we determine that this is an equation of type 3) - it clearly lacks the independent variable $x$. We take $y$ as the new argument and perform the substitution $y^{\prime}=p=p(y(x))$. By the chain rule, we find $y^{\prime \prime}=\left(y^{\prime}\right)_{x}^{\prime}=(p)_{y}^{\prime} \cdot y_{x}^{\pri... | 2\operatorname{arctg}e^{x-1} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,333 |
Example 4. Find a particular solution of the differential equation
$$
x^{4} \cdot y^{\prime \prime \prime}-2 x^{3} \cdot y^{\prime \prime}=\frac{5}{x}
$$
satisfying the initial conditions $y(1)=-\frac{1}{18}, y^{\prime}(1)=-\frac{7}{18}$, $y^{\prime \prime}(1)=\frac{1}{6}$. | Solution. The given equation belongs to type 2) with $n=3$ and $k=2$. Let $y^{\prime \prime}=p(x)$. Then $y^{\prime \prime \prime}=p^{\prime}$. After dividing the equation term by term by $x^{4}$ and making the appropriate substitutions, we arrive at the equation
$$
p^{\prime}-\frac{2}{x} p=\frac{5}{x^{5}}
$$
This is... | y_{4}=\frac{x^{4}}{12}-\frac{5}{36x^{2}}-x+1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,334 |
Example 5. Solve the equation $\frac{\left(y^{\prime \prime}\right)^{2}-y^{\prime} \cdot y^{\prime \prime \prime}}{\left(y^{\prime}\right)^{2}}=\frac{1}{x^{2}}$. | Solution. Before solving a given equation, it makes sense to analyze its structure to identify integrable expressions or combinations.
In this case, we notice the equalities (derivatives are taken with respect to $x$):
$$
\frac{\left(y^{\prime \prime}\right)^{2}-y^{\prime} \cdot y^{\prime \prime \prime}}{\left(y^{\pr... | \frac{C_{2}}{C_{1}}e^{C_{1}x}(x-\frac{1}{C_{1}})+C_{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,335 |
Example 6. Solve the differential equation $y^{\prime \prime}\left(1+2 \ln y^{\prime}\right)=1$. | Solution. The equation can be classified as both type 2) and type 3).
a) Accepting, as for type 2), \( p = y'(x), p' = y''(x) \). We arrive at the equation with separated variables \((1 + 2 \ln p) dp = dx\). After integration, we get \(2p \ln p - p = x + C_1\), or \(2y' \ln y' - y' = x + C_1\).
This equation is not s... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,336 |
Example 1. Check for linear dependence or linear independence of the following pairs of functions:
a) $y_{1}=x, y_{2}=x^{2}$
b) $y_{1}=1, y_{2}=x$
c) $y_{\mathrm{I}}=x, y_{2}=2 x$;
d) $y_{1}=\cos x, y_{2}=C \cos x$. | Solution. All functions are defined on the entire real line.
Pairs of functions a) and b) are linearly independent, as their ratios are not constant values: a) $\frac{y_{2}}{y_{1}}=x$; b) $\frac{y_{2}}{y_{1}}=x$.
Pairs of functions c) and d) are linearly dependent, as their ratios are constants: c) $\frac{y_{2}}{y_{1... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,337 |
Example 2. Check that the given functions are solutions to the given differential equations, and form the general solutions of these equations:
1) $y_{1}=3 \sin 5 x, \quad y_{2}=\frac{1}{5} \cos 5 x ; \quad y^{\prime \prime}+25 y=0 ;$
2) $y_{\mathrm{I}}=e^{3 x}, \quad y_{2}=x e^{3 x} ; \quad y^{\prime \prime}-6 y^{\pr... | Solution. 1) We find the first and second derivatives of the first function:
$$
y_{1}^{\prime}=15 \cos 5 x, \quad y_{1}^{\prime \prime}=-75 \sin 5 x
$$
Substituting these values into the given equation, we obtain the identity:
$$
y_{\mathrm{I}}^{\prime \prime}+25 y_{1}=-75 \sin 5 x+75 \sin 5 x \equiv 0, \quad x \in ... | y_{00}=\widetilde{C}_{1}\sin5x+\widetilde{C}_{2}\cos5x,\quady_{00}=C_{1}e^{3x}+C_{2}\cdotxe^{3x},\quady_{00}=C_{1}e^{2x}+C_{2}e^{-3x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,338 |
Example 3. Find the general solutions of the given linear homogeneous differential equations with constant coefficients. In those cases where initial conditions are given, find the corresponding particular solutions:
1) $2 y^{\prime \prime}-3 y^{\prime}+y=0$
2) $9 y^{\prime \prime}+12 y^{\prime}+4 y=0 ; x=1, y=2, y^{\... | Solution. All differential equations are solved according to a unified scheme, so minor details are sometimes omitted. We form the characteristic equation, find its roots, and construct the general solution of the homogeneous linear differential equation (LODE) according to the algorithm in point $4^{\circ}$.
1) We ha... | y_{00}=C_{1}e^{x}+C_{2}e^{0.5x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,339 |
Example 1. Solve the equation $y^{\prime \prime}+y^{\prime}-2 y=x^{3}-2$. | Solution. We have: $k^{2}+k-2=0, k_{1}=-2, k_{2}=1, y_{00}=C_{1} e^{-2 x}+$ $+C_{2} e^{x}$. The right-hand side of the non-homogeneous equation is $P_{n}(x) e^{x}=$ $=\left(x^{3}-2\right) e^{0 x}$, i.e., $n=3, \lambda=0$ is not a root of the characteristic equation. According to point 1) of theorem 3, a particular solu... | y_{\text{on}}=C_{1}e^{-2x}+C_{2}e^{x}-\frac{x^{3}}{2}-\frac{3x^{2}}{4}-\frac{9x}{4}-\frac{7}{8} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,340 |
Example 2. Find the general solution of the equation $y^{\prime \prime}+y^{\prime}=x^{2}+2 x$. Also, specify a particular solution that satisfies the initial conditions $x=0, \quad y=1, \quad y^{\prime}=-3$. | Solution. We have $k^{2}+k=0, k_{1}=0, k_{2}=-1, y_{00}=C_{1}+C_{2} e^{-x}$. The right-hand side has the form $P_{n}(x) e^{x}=\left(x^{2}+2 x\right) e^{0 x}$, i.e., $n=2, \lambda=0=k_{1}$ - a single root of the characteristic equation. According to point 2) of theorem 3:
0: $y_{4}=x\left(A x^{2}+B x+C\right)=A x^{3}+B... | y_{4}=-2+3e^{-x}+\frac{x^{3}}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,341 |
Example 3. Solve the equation $y^{\prime \prime}+2 y^{\prime}+y=(1+x) e^{-x}$ | Solution. We have: $k^{2}+2 k+1=0, k_{\mathrm{I}}=k_{2}=-1, y_{\text {gen }}=\left(C_{\mathrm{I}}+\right.$ $\left.+C_{2} x\right) e^{-x}$. Given $f(x)=(1+x) e^{-x}, \lambda=-1-$ is a double root of the characteristic equation, $n=1$. According to point 3) of Theorem 3, we will form the corresponding system, preliminari... | y_{\text{gen}}=(C_{1}+C_{2}x)e^{-x}+(\frac{x^{3}}{6}+\frac{x^{2}}{2})e^{-x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,342 |
Example 4. Solve the equation $y^{\prime \prime}+9 y=e^{-x}(-\cos 3 x+$ $+(x+2) \sin 3 x)$. | Solution. $k^{2}+9=0, k= \pm 3 i$. Compare the right-hand side of the equation with $f(x)$ from Theorem 8. We have $P_{n}(x)=-1, n=0, Q_{m}(x)=x+2, m=1$, $s=\max (0,1)=1$, the numbers $\gamma \pm i \delta=-1 \pm 3 i$ are not roots of the characteristic equation. According to Theorem 4, point 1, we will have (when diffe... | y_{\text{p}}=C_{1}\cos3x+C_{2}\sin3x+[(\frac{6}{37}x+\frac{641}{1369})\cos3x+(\frac{1}{37}x+\frac{298}{1369})\sin3x]e^{-x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,343 |
Example 5. Solve the equation $y^{\prime \prime}+4 y=-8 \sin 2 x+32 \cos 2 x+$ $+4 e^{2 x}$. | Solution. $k^{2}+4=0, k_{1,2}= \pm 2 i$. Next (see Theorem 2) we find two particular solutions (solve as if two equations, which is equivalent to one equation with two independent right-hand sides).
a) We find the particular solution corresponding to the trigonometric part, keeping in mind that the numbers $\gamma \pm... | y_{ | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,344 |
Example 6. Solve the equation $y^{\prime \prime}+6 y^{\prime}+8 y=\frac{4 e^{-2 x}}{2+e^{2 x}}$. | Solution. We have: $k^{2}+6 k+8=0, k_{1}=-4, k_{2}=-2, y_{o 0}=$ $=C_{1} e^{-4 x}+C_{2} e^{-2 x}$. We take $y_{1}=e^{-4 x}, \quad y_{2}=e^{-2 x}$ as two linearly independent particular solutions of the homogeneous linear differential equation. Then $y_{1}^{\prime}=-4 e^{-4 x}$, $y_{2}^{\prime}=-2 e^{-2 x}$ and we solve... | y_{\text{on}}=C_{1}e^{-4x}+C_{2}e^{-2x}-e^{-4x}\ln(2+e^{2x})+\frac{1}{2}(2x-\ln(2+e^{2x}))e^{-2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,345 |
Example 1. Find the general solution of the differential equation
$$
y^{\prime \prime \prime}-4 y^{\prime \prime}+7 y^{\prime}-4 y=0
$$ | Solution. The roots of the characteristic equation $k^{3}-4 k^{2}+$ $+7 k-4=0$ can be determined by the method of factorization: $\left(k^{2}-3 k+4\right)(k-1)=0$.
We find $k_{1}=1, k_{2}=\frac{3-\sqrt{7} i}{2}, k_{2}=\frac{3+\sqrt{7} i}{2}$.
Answer. $y_{00}=C_{1} e^{x}+\left(C_{2} \cos \frac{\sqrt{7}}{2} x+C_{3} \si... | y_{00}=C_{1}e^{x}+(C_{2}\cos\frac{\sqrt{7}}{2}x+C_{3}\sin\frac{\sqrt{7}}{2}x)e^{(3/2)x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,346 |
Example 3. Solve the differential equation $y^{\prime \prime \prime}+y^{\prime}=\frac{\cos x}{\sin ^{2} x}$. | Solution. The roots of the characteristic equation $k^{3}+k=0$ are the numbers $k_{1}=0, k_{2}=i, k_{3}=-i$. Therefore, the general solution of the homogeneous equation has the form $y_{00}=C_{1}+C_{2} \cos x+C_{3} \sin x$. A particular solution of the non-homogeneous equation can only be determined using the method of... | y_{\text{on}}=C_{1}+C_{2}\cosx+C_{3}\sinx+\frac{1}{\sinx}+(\operatorname{ctg}x+x)\cosx+\sinx\ln|\sinx| | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,347 |
Example 1. Solve the system
$$
\left\{\begin{array}{l}
y_{1}^{\prime}=y_{2}+y_{3} \\
y_{2}^{\prime}=y_{1}+y_{3} \\
y_{3}^{\prime}=y_{1}+y_{2}
\end{array}\right.
$$
Find the particular solution if $x=0, y_{1}=1, y_{2}=2, y_{3}=0$. | Solution. We are dealing with a normal and linear system. Let's use the scheme of point $2^{\circ}$.
l) $y_{1}^{\prime \prime}=y_{2}^{\prime}+y_{3}^{\prime}=2 y_{1}+y_{2}+y_{3}=2 y_{1}+y_{1}^{\prime}$.
We differentiated the first equation (with respect to the variable $x$, considered the argument of all three unknown ... | y_{1}=e^{2x},y_{2}=e^{-x}+e^{2x},y_{3}=-e^{-x}+e^{2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,348 |
Example 2. Solve the system of equations $\left\{\begin{array}{l}x^{\prime}=-4 x-14 y, \\ y^{\prime}=5 x+13 y .\end{array}\right.$ | Solution. Immediately note that the unknown functions are denoted by different letters, and we will consider the variable $t$ as the argument.
The essence of the method is to transition from the given system to another, equivalent system, in which differential groups or complete differentials can be separated. For exa... | {\begin{pmatrix}\frac{2}{3}C_{1}e^{3}+C_{2}e^{6}\\-\frac{1}{3}C_{1}e^{-3}-\frac{5}{7}C_{2}e^{6}\end{pmatrix}.} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,349 |
Example 3. Solve the system of equations
$$
\left\{\begin{array}{l}
x^{\prime}=3 x-y+z \\
y^{\prime}=-x+5 y-z \\
z^{\prime}=x-y+3 z
\end{array}\right.
$$ | Solution. We will reduce the given system to a single third-order equation using the method described above in point $2^{\circ}$. The order of using one or another equation can be arbitrary.
1) Differentiate (with respect to $t$) the first equation of the system: $x^{\prime \prime}=3 x^{\prime}-$ $-y^{\prime}+z^{\prim... | {\begin{pmatrix}C_{1}e^{2}&+C_{2}e^{3}+C_{3}e^{6}\\&C_{2}e^{3}-2C_{3}e^{6}\\-C_{1}e^{2}+C_{2}e^{3}+C_{3}e^{6}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,350 |
Example 4. Solve the system $\left\{\begin{array}{l}x^{\prime}=2 x-4 y, \\ y^{\prime}=x-3 y .\end{array}\right.$. Also find a particular solution of the system that satisfies the initial conditions $t=0, x=1$, $y=2$. | Solution. From the system, we obtain one second-order equation with one unknown function. First, we differentiate the first equation (with respect to $t$) and substitute the expression for $y^{\prime}$ from the second equation: $x^{\prime \prime}=2 x^{\prime}-4 y^{\prime}$, where $y^{\prime}=x-3 y$. We get $x^{\prime \... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,351 | |
Example 5. Solve the system of differential equations
$$
\left\{\begin{array}{l}
x^{\prime}=2 x-4 y \\
y^{\prime}=x-3 y
\end{array}\right.
$$ | Solution. Let $A=\left(\begin{array}{cc}2 & -4 \\ 1 & -3\end{array}\right)$ be the matrix composed of the coefficients from the right-hand side of the system. We write down the characteristic equation of matrix $A$:
$$
\left|\begin{array}{cc}
2-\lambda & -4 \\
1 & -3-\lambda
\end{array}\right|=-6+3 \lambda-2 \lambda+\... | {\begin{pmatrix}4B_{1}e^{}+B_{2}e^{-2},\\B_{1}e^{}+B_{2}e^{-2}0\end{pmatrix}.} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,352 |
Example 6. Solve the system of differential equations
$$
\left\{\begin{array}{l}
x^{\prime}=3 x+4 y+2 z \\
y^{\prime}=x+4 y+z \\
z^{\prime}=4 x+6 y+5 z
\end{array}\right.
$$ | Solution. The characteristic equation of the matrix $A=\left(\begin{array}{lll}3 & 4 & 2 \\ 1 & 4 & 1 \\ 4 & 6 & 5\end{array}\right)$ is: $\left|\begin{array}{ccc}3-\lambda & 4 & 2 \\ 1 & 4-\lambda & 1 \\ 4 & 6 & 5-\lambda\end{array}\right|=0$, or $(3-\lambda)(4-\lambda) \times$ $\times(5-\lambda)+12+16-6(3-\lambda)-4(... | {\begin{pmatrix}C_{1}e^{}&+7C_{3}e^{9}\\&C_{2}e^{2}+4C_{3}e^{9}\\-C_{1}e^{}-2C_{2}e^{2}+13C_{3}e^{9}\end{pmatrix}.} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,353 |
Example 7. Solve the system $\left\{\begin{array}{l}x^{\prime}=3 x-y+z, \\ y^{\prime}=-x+5 y-z, \\ z^{\prime}=x-y+3 z .\end{array}\right.$ | Solution. Let's provide a brief schematic solution to reinforce the methodology of eigenvalues and eigenvectors.
$$
A=\left(\begin{array}{ccc}
3 & -1 & 1 \\
-1 & 5 & -1 \\
1 & -1 & 3
\end{array}\right)
$$
1) Characteristic equation of the matrix
$A=\left|\begin{array}{ccc}3-\lambda & -1 & 1 \\ -1 & 5-\lambda & -1 \\... | (c_1e^{2t}(-1,0,1)+c_2e^{3t}(1,1,1)+c_3e^{6t}(-1,2,-1)) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,354 |
Example 8. Integrate the system of differential equations $\left\{\begin{array}{l}x^{\prime}=\frac{y+e^{x}}{y+e^{t}}, \\ y^{\prime}=\frac{y^{2}-e^{x+t}}{y+e^{t}} .\end{array}\right.$ | Solution. In this system, the unknown functions $x$ and $y$ depend on the argument $t$.
1) Multiply both sides of the second equation by $e^{-t}$:
$$
e^{-t} y^{\prime}=\frac{y^{2} e^{-t}-e^{x}}{y+e^{t}}
$$
Add this equality to the first equation of the system and to the identity $-e^{-t} y=-e^{-t} y$. We get:
$$
e^... | {\begin{pmatrix}e^{-}y+C_{1},\\e^{-x}y+=C_{2}0\end{pmatrix}.} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,355 |
Example 9. Integrate the system of equations $\left\{\begin{array}{l}x^{\prime}=\frac{y^{3}}{x^{2}}, \\ y^{\prime}=\frac{x^{3}}{y^{2}}\end{array}\right.$ | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,356 | |
Example 1. Investigate the convergence of the series $\sum_{n=2}^{\infty} \frac{18}{n^{2}+n-2}$ and, if possible, find its sum. | Solution. We have $\frac{18}{n^{2}+n-2}=\frac{6}{n-1}-\frac{6}{n+2}$. Let's form the partial sum and find its limit. Notice which terms cancel each other out (!). We have:
$$
\begin{aligned}
& S_{n}=u_{2}+u_{3}+\ldots+u_{n}= \\
& =6\left[\left(\frac{1}{1}-\frac{1}{A}\right)+\left(\frac{1}{2}-\frac{1}{b}\right)+\left(\... | 11 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,357 |
Example 2. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$.
翻译结果：
Example 2. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$. | Solution. In fact, we are dealing with a Dirichlet series with $p=\frac{1}{2}$, which, as noted above, diverges. Nevertheless, it makes sense to indicate the possibility of estimating the partial sum of this series from below. We have:
$S_{n}=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n}}>\underbrac... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,358 |
Example 3. Investigate the convergence of the series $\sum_{n=2}^{\infty} \frac{n-1}{3 n+20}$. | Solution. Let's check the necessary condition for convergence (Theorem 1). We have \(\lim _{n \rightarrow+\infty} u_{n}=\lim _{n \rightarrow \infty} \frac{n-1}{3 n+20}=\lim _{n \rightarrow \infty} \frac{n\left(1-\frac{1}{20}\right)}{n\left(3+\frac{20}{n}\right)}=\frac{1}{3} \neq 0\).
The series diverges. Since its ter... | \cdot | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,359 |
Example 7. Investigate the convergence of the series $\sum_{n=1}^{\infty}\left(\frac{n+1}{3 n-1}\right)^{n^{2}}$. | Solution. We apply the Cauchy radical test. We have:
$$
q=\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}=\lim _{n \rightarrow \infty}\left(\frac{n+1}{3 n-1}\right)^{n}=\lim _{n \rightarrow \infty}\left(\frac{1+\frac{1}{n}}{3-\frac{1}{n}}\right)^{n}=0
$$
Answer. The series converges. | Theseriesconverges | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,361 |
Example 8. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^{2}+1}$. | Solution. We propose to verify that the D'Alembert and Cauchy (radical) criteria are not applicable, as in this case \( q=1 \). We will apply the Cauchy integral test.
\[
\begin{aligned}
& \text { We have } a_{n}=\frac{1}{n^{2}+1} . \text { Let } f(x)=\frac{1}{x^{2}+1} . \text { Then } f(n)=a_{n} \text { and } \\
& \i... | \frac{\pi}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,362 |
Example 9. Investigate the convergence of the series $\sum_{n=2}^{\infty} \frac{1}{n \ln n}$. | Solution. We have $\int_{2}^{+\infty} \frac{d x}{x \ln x}=\left.\lim _{A \rightarrow+\infty} \ln |\ln x|\right|_{2} ^{A}=+\infty$.
Answer. The series diverges. | Theseriesdiverges | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,363 |
Example 10. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{n^{2}+7}{n^{3}+3 n-2}$ | Solution. Note that the necessary condition for convergence is satisfied:
$$
\lim _{n \rightarrow \infty} \frac{n^{2}+7}{n^{3}+3 n-2}=\lim _{n \rightarrow \infty} \frac{1+\frac{7}{n^{2}}}{1+\frac{3}{n^{2}}-\frac{2}{n^{3}}}=0
$$
However, this does not ensure the convergence of the series.
D'Alembert's ratio test:
$$... | Theseriesdiverges | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,364 |
Example 11. Investigate the convergence of the series $\sum_{n=1}^{\infty} \ln \left(1+\frac{5}{n^{2}}\right)$. | Solution. We take the convergent series $\sum_{n=1}^{\infty} \frac{1}{n^{2}}$ as a comparison series. Since $\lim _{n \rightarrow \infty} \frac{\ln \left(1+\frac{5}{n^{2}}\right)}{\frac{1}{n^{2}}}=5$ (a consequence of the second remarkable limit), the series under investigation converges.
Answer. The series converges.... | Theseriesconverges | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,365 |
Example 1. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{\cos ^{3} n \alpha}{n^{2}}$. | Solution. The given series is alternating, since $\cos ^{3} n \alpha>0$ for some $n \in N$, while for other $n$ we have $\cos ^{3} n \alpha<0$. The series of absolute values of the terms of the given series converges by comparison theorem 5:
$$
a_{n}=\left|\frac{\cos ^{3} n \alpha}{n^{2}}\right| \leqslant \frac{1}{n^{... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,366 |
Example 2. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt[3]{n+2}}$. | Solution. The series $\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n+2}}$, formed from the absolute values of the given series, diverges (it is comparable to the divergent Dirichlet series $\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n}}\left(p=\frac{1}{3}\right): \frac{1}{\sqrt[3]{n+2}} \approx \frac{1}{\sqrt[3]{n}}(n \rightarrow ... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,367 |
Example 1. Investigate the convergence of the functional series $\sum_{n=1}^{\infty} \frac{1}{n^{2}+3} \cdot\left(\frac{x+1}{x-1}\right)^{n}$ | Solution. Let $q=q(x)=\frac{x+1}{x-1}$. The series $\sum_{n=1}^{\infty} \frac{q^{n}}{n^{2}+3}$ converges absolutely when $|q| \leqslant 1$ by the comparison theorem $5 .\left(\left|\frac{q^{n}}{n^{2}+3}\right| \leqslant \frac{1}{n^{2}+3}\right.$, if $|q|1$, then the series $\sum_{n=1}^{\infty} \frac{q^{n}}{n^{2}+3}$ di... | x\leqslant0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,369 |
Example 3. Investigate the convergence of the power series $\sum_{n=0}^{\infty} \frac{x^{n}}{n!}$. | Solution. Let's determine the radius of convergence of the power series using D'Alembert's formula. We have (it is customary to consider $0!=1$):
$$
R=\lim _{n \rightarrow \infty}\left|\frac{a_{n}}{a_{n+1}}\right|=\lim _{n \rightarrow \infty} \frac{(n+1)!}{n!}=\lim _{n \rightarrow \infty}(n+1)=+\infty
$$
Answer. The ... | Thegivenseriesconvergesabsolutelyforallx\in\mathbb{R} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,370 |
Example 4. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{2^{n}(x+1)^{n}}{\left(3^{n}+2\right) n}$ | Solution. Let $y=x+1$ and investigate the power series $\sum_{n=1}^{\infty} \frac{2^{n} y^{n}}{\left(3^{n}+2\right) n}$
1) $R=\lim _{n \rightarrow \infty} \frac{\left|a_{n}\right|}{\left|a_{n+1}\right|}=\lim _{n \rightarrow \infty} \frac{2^{n}}{\left(3^{n}+2\right) n} \cdot \frac{\left(3^{n+1}+2\right)(n+1)}{2^{n+1}}=... | Thegivenseriesconvergesabsolutelyintheinterval(-\frac{5}{2};\frac{1}{2}),convergesconditionallyat-\frac{5}{2},divergeswhenx<-\frac{5}{2}x\geqslant\frac{1}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,371 |
Example 5. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{x^{2 n-1}}{\left(n^{2}+1\right) \cdot 3^{n}}$. | Solution. For this series, the formulas of D'Alembert and Cauchy are not applicable - the corresponding limits do not exist (the coefficients of the series at even powers of $x$ are zero, while the coefficients at odd powers are non-zero; thus, in particular, $\frac{a_{2 n}}{a_{2 n+1}}=0$, and $\frac{a_{2 n+1}}{a_{2 n+... | -\sqrt{3}\leqslantx\leqslant\sqrt{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,372 |
Example 1. Expand the function $f(x)=\frac{1}{1-x}$ into a Taylor series in powers of: a) $x-1$, b) $x$, c) $x+1$. | Solution. a) Expansion in a series of powers of $(x-1)$ is impossible, since $f(x)$ is not defined at the point $x=1$. In other words, a series of powers of $(x-1)$ for this function does not exist.
Cases b) and c) will be considered together. Expansion in powers of $x$ and $(x+1)$ is possible, because $f(x)$ is infin... | \begin{aligned}&b)\frac{1}{1-x}=1+x+x^{2}+x^{3}+\ldots+x^{n}+\ldots\quad(|x|<1)\\&)\frac{1}{1-x}=\frac{1}{2}+\frac{1}{2^{2}}(x+1)+\frac{1}{2^{3}}(x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,373 |
Example 2. Expand the function $f(x)=\arcsin x$ into a Maclaurin series. | Solution. Substitution in (4) $\alpha=-\frac{1}{2}$ and replacement of $x$ with $-x^{2}$ leads to the expansion
$$
\begin{aligned}
& \frac{1}{\sqrt{1-x^{2}}}=1+\frac{1}{2} x^{2}+\frac{1 \cdot 3}{2^{2} \cdot 2!} x^{4}+\frac{1 \cdot 3 \cdot 5}{2^{3} \cdot 3!} x^{6}+\ldots+ \\
& \quad+\frac{1 \cdot 3 \cdot 5 \cdot \ldots... | \arcsinx+\frac{x^{3}}{2\cdot3}+\frac{1\cdot3}{2^{2}\cdot2!}\cdot\frac{x^{5}}{5}+\frac{1\cdot3\cdot5}{2^{3}\cdot3!}\cdot\frac{x^{7}}{7}+\ldots+\frac{1\cdot3\cdot5} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,374 |
Example 3. Expand the function $f(x)=\ln \frac{1+x}{1-x}$ into a Maclaurin series and, using this expansion, compute $\ln 3$ with an accuracy of $10^{-3}$. | Solution. We will use the expansion from part b) of Example 1:
$$
\frac{1}{1-t}=1+t+t^{2}+t^{3}+\ldots+t^{n}+\ldots
$$
Replace \( t \) with \( t^2 \) in this expansion. We get
$$
\frac{1}{1-t^{2}}=1+t^{2}+t^{4}+t^{6}+\ldots+t^{2 n}+\ldots \quad, \quad(|t|<1)
$$
Integrate this equality over the interval \([0 ; x]\) ... | 1.098\10^{-3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,375 |
Example 4. Calculate with an accuracy of $10^{-3}$ the value of the definite integral $\int_{0}^{1} \frac{e^{-x^{2}}-1}{x} d x$. | Solution. The given integral, as well as integrals of the form $\int_{a}^{b} \frac{\sin x}{x} d x$, $\int_{a}^{b} \frac{\cos x}{x} d x$, and some others, cannot be evaluated using the Newton-Leibniz formula. The reason is that the antiderivatives of such functions cannot be expressed in terms of elementary functions. T... | 0.659 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,376 |
Example 5. Find the sum of the series $f(x)=\sum_{n=0}^{\infty}\left(n^{2}+3 n-5\right) \cdot x^{n}$. | Solution. First, let's determine where the given series converges. Using D'Alembert's formula, it is easy to establish (do this) that the radius of convergence of the series is $R=1$. Therefore, the sum of the series is defined in the interval $(-1 ; 1)$.
The given series can be represented as the sum of three series:... | f(x)=\frac{7x^{2}-14x+5}{(x-1)^{3}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,377 |
Example 6. Write several terms of the expansion of the function $y=\operatorname{tg} x$ in a series of powers:
a) $x ; \quad$ b) $x-\frac{\pi}{4}$. | Solution. It is not possible to use any standard expansions or formulas. Therefore, the coefficients of the series will be calculated using the general formulas of Taylor-Maclaurin (see Theorem 18). We have
$$
\begin{gathered}
y=\operatorname{tg} x, \quad y^{\prime}=\frac{1}{\cos ^{2} x}, \quad y^{\prime \prime}=\frac... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,378 |
Example 1. Suppose that $f(x)$ is a function continuous on the entire real line and periodic with period $2 \pi$. Derive the formulas for computing the coefficients of its Fourier series. | Solution. Suppose the given function is represented by a trigonometric series that converges uniformly on the entire line, in particular on the interval $[-\pi ; \pi]$, and write this series as: $f(x)=\frac{\alpha_{0}}{2}+\alpha_{\mathrm{I}} \cos x+\beta_{\mathrm{I}} \sin x+$
$$
+\alpha_{2} \cos 2 x+\beta_{2} \sin 2 x... | \alpha_{0}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x),\alpha_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cosnx,\beta_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sinnx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,379 |
Example 4. Expand the function $f(x)=10-x$, $5<x<15$ into a Fourier series. | Solution. We will construct the graph of the function $f(x)$ (Fig. 5.3). The dashed line represents the periodic extension with a period of $2l=10$ $(l=5)$. From the drawing, we conclude that the Fourier series expansion of this function coincides with the expansion of the function $f(x)=-x, x \in (-5; 5)$. This allows... | 10-\frac{10}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\sin\frac{n\pix}{5}(5<x<15) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,381 |
Task 1 (scheduling problem). In how many ways can a daily school schedule be made, consisting of 6 subjects out of 10? | Solution. If we take, for example, BAGRIF (biology, algebra, geometry, Russian, history, physical education) as one of the possible schedules, then any other schedule must differ from it either in the order of subjects or in the subjects themselves, i.e., we are talking about permutations of six different subjects out ... | 151200 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,383 |
Problem 4 (Permutation Problem). In how many ways can 13 different math books be arranged on a bookshelf?
1) in any order;
2) so that 4 books on arithmetic are together (in any order);
3) so that 4 books on arithmetic are together (in any order) and 6 books on algebra are together (in any order);
4) so that 4 books on... | Solution. 1) The number of ways to arrange 13 books in any order is equal to the number of permutations of 13 elements (objects): $P_{\mathrm{I} 3}=13!=6227020800$.
2) Imagine that the 4 books on arithmetic are tied together. Then this bundle (package) can be considered as one item, which means we need to arrange 10 i... | 6227020800;87091200;2073600;622080 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,384 |
Problem 5 (Partition problem) In how many ways can 11 items be divided into two groups such that each group has no fewer than three items (selecting one group uniquely determines the other)? | Solution. All possibilities to divide 11 items into two groups are represented by the following decompositions: $11=3+8, 11=4+7$, $11=5+6$ (the possibilities $11=6+5, 11=7+4$ and $11=8+3$ coincide with the previous ones). It remains to understand in how many ways one can choose 3, 4, or 5 items out of 11. Since the ord... | 957 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,385 |
Task 6 (the problem of exiting a vehicle). In an elevator on the first floor of a twelve-story building, 6 passengers entered. In how many ways can they exit the elevator on the required floors, starting from the third? | Solution. For one of the passengers, there are 10 possibilities to exit the elevator on one of the 10 floors (from the third to the twelfth inclusive). The other passenger also has 10 possibilities to exit the elevator. The same can be said about the other passengers. Thus, the multiplication rule applies, according to... | 1,000,000 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,386 |
Problem 1. In an urn, there are two red, five blue, and three white balls. Find the probability that two balls drawn simultaneously will be blue (event $A$). | Solution. There are a total of 10 balls in the urn. The total number of all elementary outcomes is equal to the number of ways to draw 2 balls out of 10: $n=C_{10}^{2}=\frac{10 \cdot 9}{1 \cdot 2}=45$. The number of outcomes favorable to event $A$ is equal to the number of ways to draw 2 balls out of 5, i.e., $m=C_{5}^... | p(A)=\frac{2}{9} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,387 |
Problem 4. In an urn, there are 10 balls, among which 2 are red, 5 are blue, and 3 are white. Find the probability that a randomly drawn ball will be colored (event $A$). | Solution. The following outcomes are possible: $E_{1}$ - a red ball is drawn, $E_{2}$ - a blue ball is drawn, $E_{3}$ - a white ball is drawn. The number of all possible outcomes is equal to the number of balls, i.e., ten, $n=10$. Of these, the number of outcomes $E_{1}$ is two, $E_{2}$ is five, and $E_{3}$ is three.
... | 0.7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,388 |
Problem 5. Three balls are drawn at once from the urn (condition of problem 4). Find the probability that all the drawn balls will be blue (event $B$). | Solution. The number of possible outcomes $n=C_{10}^{3}$, and the number of outcomes favorable to event $B_{1}, m=C_{5}^{3}$, since three balls are drawn from five blue ones, therefore $p(B)=\frac{C_{5}^{3}}{C_{10}^{3}}=\frac{\frac{5 \cdot 4 \cdot 3}{1 \cdot 2 \cdot 3}}{\frac{10 \cdot 9 \cdot 8}{1 \cdot 2 \cdot 3}}=\fr... | \frac{1}{12} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,389 |
Problem 6. In a batch of $N$ items, there are $M(M \leqslant N)$ marked items. $n(n \leqslant N)$ items are randomly selected from the given batch. Find the probability that among these $n$ items, there will be exactly $m$ $(m \leqslant n, m \leqslant M$) marked items (event $A$). | Solution. The total number of cases in which event $A$ can occur is $n_{0}=C_{N}^{n}$.
To count the number of cases favorable to event $A$, consider a sample of $n$ items, among which exactly $m$ are marked. The total number of outcomes in which $m$ items out of $M$ can enter the sample is $C_{M}^{m}$. Each such combi... | \frac{1}{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,390 |
Problem 9 (Meeting Problem). Two students agreed to meet at a certain place between 12:00 and 13:00. The one who arrives first waits for the other for $\alpha$ minutes ( $\alpha<60$ ), after which he leaves. What is the probability of their meeting (event $A$), if each of them can arrive at any time during the specifie... | Solution. Let the arrival time (counting from 12 o'clock) of one of the students be denoted by $x$, and the other student's arrival time by $y$. For the meeting to occur, it is necessary and sufficient that the inequality $|x-y| \leqslant \alpha$ holds, which is equivalent to the following: $-\alpha \leqslant x-y \leqs... | \frac{11}{36} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,391 |
Problem 10. Among the first 4000 numbers of the natural number sequence, there are 551 prime numbers. Find the relative frequency of the occurrence of a prime number (event $A$). | Solution. By formula (3) we have: $p(A)=\frac{551}{4000}=0.13775$.
Answer. 0.138 . | 0.138 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 31,392 |
Task 11. Ten different books are randomly arranged on one bookshelf. Find the probability that two specific books will be placed next to each other (event $A$). | Solution. The number $n$ of all elementary outcomes of the experiment in which event $A$ can occur is $C_{10}^{2}$, and the number $m$ of outcomes favorable to event $A$ is 9, since the two specific books will be placed next to each other if they occupy either the first-second positions, the second-third, and so on, up... | 0.2 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,393 |
Problem 1. The probability that a shooter, upon firing a shot, will score 10 points is 0.4; 9 points - 0.3; and finally, 8 or fewer points - 0.3. Find the probability that the shooter will score no less than 9 points in one shot. | Solution. Let's denote the events and write down the corresponding probabilities:
$A$ - hitting $10, p(A)=0.4$;
$B$ - hitting $9, p(B)=0.3$;
$C$ - hitting 8 or less, $p(C)=0.3$.
$A+B-$ hitting either 10 or $9, p(A+B)=$ ?
Since events $A, B, C$ are mutually exclusive, by formula (6) we have:
$$
p(A+B)=p(A)+p(B)=0.... | 0.7 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,394 |
Problem 3. In an urn, there are 30 balls: 10 red, 5 blue, and 15 white. What is the probability of drawing a colored ball if one ball is drawn? | Solution. The probability of drawing a red ball $p(A)=\frac{10}{30}$, blue $p(B)=\frac{5}{30}$. The probability of drawing a colored ball $p(A+B)=p(A)+p(B)=\frac{1}{2}$
## Exercises
1. In an urn, there are balls of four colors: white balls $50 \%$, red $20 \%$, green $20 \%$, blue $10 \%$. What is the probability tha... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,396 |
Example 2. In an urn, there are 7 white and 3 black balls. The condition of the experiment is that each drawn ball is returned to the urn. Event $A$: a white ball is drawn in the first trial; event $B$: a white ball is drawn in the second trial. We have $p(A)=p(B)=\frac{7}{10}$ (if the first ball was black, $p(B)$ woul... | Solution. Consider independent events: $A$ - drawing a white ball from the 1st urn; $B$ - drawing a white ball from the 2nd urn. We are interested in the event $A B$. We have $p(A)=\frac{2}{12}, p(B)=\frac{8}{12}$. By formula (10) we get $p(A B)=p(A) p(B)=\frac{1}{6} \cdot \frac{2}{3}=\frac{1}{9}$. | \frac{1}{9} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,397 |
Problem 2. The probability of a student passing the credit is 0.8. If the credit is passed, the student is allowed to take the exam, the probability of passing which is 0.9. Find the probability that the student will pass the credit and the exam. | Solution. Let's denote the events: $A$ - passing the test, $B$ - passing the exam. Then $A B$ means that the student passes both the test and the exam. Event $B$ depends on event $A$, so $p(A)=0.8, p(B / A)=0.9$. By formula (9) we have $p(A B)=0.8 \cdot 0.9=0.72$. | 0.72 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,398 |
Problem 3. In an urn, there are $a$ white and $b$ black balls. What is the probability that one of the two balls drawn is white and the other is black? (The drawn ball is not returned to the urn). | Solution. Let's denote the events:
$A$ - a white ball appears during the first draw;
$B$ - a black ball appears during the second draw;
$C$ - a black ball appears during the first draw;
## $D$ - a white ball appears during the second draw.
Then:
$A B$ - the first ball drawn is white, and the second is black;
$C ... | \frac{2ab}{(+b)(+b-1)} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,399 |
Problem 6. The probability of the first shooter hitting the target is $p_{1}$, and the second shooter is $p_{2}$. The shooters fired simultaneously. What is the probability that:
a) both shooters hit the target;
b) only one hits;
c) at least one hits? | Solution. Let's denote the events:
$A$ - the first shooter hits; $p(A)=p_{1}$; denote $q_{1}=p(\bar{A})=$ $=1-p_{1}$
$B$ - the second shooter hits; $p(B)=p_{2}$; denote $q_{2}=p(\bar{B})=$ $=1-p_{2}$.
Then:
$A B$ - both shooters hit;
$A \bar{B}+\bar{A} B$ - one shooter hits;
$A+B$ - at least one shooter hits.
Ev... | )p_{1}p_{2},\,b)p_{1}+p_{2}-2p_{1}p_{2},\,)p_{1}+p_{2}-p_{1}p_{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,400 |
Task 7. A subscriber has forgotten the last digit of the phone number and therefore dials it at random. Determine the probability that he will need to call no more than three times to get the correct connection. | Solution. Let's denote the events:
$A_{1}$ - the subscriber got through on the first attempt;
$A_{2}$ - the subscriber got through on the second attempt;
$A_{3}$ - the subscriber got through on the third attempt.
Then $A=A_{1}+\overline{A_{1}} A_{2}+\overline{A_{1}} \overline{A_{2}} A_{3}$ - the subscriber used no ... | 0.3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,401 |
Problem 8. In 28 examination tickets, each includes two theoretical questions and one problem. The student has prepared 50 theoretical questions and can solve the problems in 22 tickets. What is the probability that, by randomly picking one ticket, the student will be able to answer all the questions in the ticket? | Solution. Consider the events:
$A_{1}$ - the student knows the first question;
$A^{2}$ - the student knows the second question;
$A_{3}$ - the student will solve the problem;
$A=A_{1} A_{2}$ - the student knows both questions;
Then $A A_{3}$ - the student will answer all the questions on the ticket. Events $A_{1}$ ... | 0.625 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,402 |
Task 1. Two shooters fired one shot each at a target. The probability of hitting the target for the first shooter is 0.7; for the second shooter, it is 0.8. Find the probability that at least one shooter will hit the target. | Solution. Let's denote the events:
$A_{1}$ - the first shooter hits the target;
$A_{2}$ - the second shooter hits the target;
Then $A_{1}+A_{2}$ - at least one shooter hits the target. Events $A_{1}$ and $A_{2}$ are compatible and independent, so by formula (15) we have:
$$
p\left(A_{\mathrm{I}}+A_{2}\right)=p\left... | 0.94 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,403 |
Problem 3. An inspector checks items for compliance with the standard. It is known that the probability of an item complying with the standard is 0.95. Find the probability that: a) both of the two inspected items will be standard, if the events of items complying with the standard are independent; b) only one of the t... | Solution. a) Let $A_{1}$ and $A_{2}$ be events indicating that the first and second items are standard, respectively. Then $p\left(A_{1}\right)=p\left(A_{2}\right)$.
Therefore, $p\left(A_{1} A_{2}\right)=p\left(A_{1}\right) p\left(A_{2}\right)=0.95 \cdot 0.95=0.9025$ b) If $B$ is the event indicating that only one of ... | )0.9025;b)0.095 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,404 |
Problem 4. There are two identical urns. The first urn contains 3 black and 7 white balls, while the second urn contains 4 black and 6 white balls. One urn is chosen at random, and a ball is drawn from it, which turns out to be white. What is the probability that this ball was drawn from the first urn? | Solution. Consider the events:
$A$ - the selected ball turned out to be white;
$H_{1}$ - the ball was drawn from the first urn;
$H_{2}$ - the ball was drawn from the second urn.
Since event $A$ can only occur with the realization of one of the two hypotheses, we will again use formula (**).
Let's calculate the pro... | 0.190,0.333,0.476 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,406 |
Task 1. In an urn, there are 8 balls, 5 of which are white. A ball is drawn from the urn 8 times, and after recording its color, it is returned to the urn. Find the probability that the white color was recorded 3 times. | Solution. Let event $A$ be the appearance of a white ball in one of the 8 draws. Then $p=5 / 8=0.625 ; q=1-p=0.375 ; n=8 ; k=3$. We need to calculate $p_{8}(3)$. According to the Bernoulli formula (19):
$$
p_{8}(3)=C_{8}^{3} \cdot 0.625^{3} \cdot 0.375^{5}=56 \cdot 0.244 \cdot 0.007=0.101
$$
Answer. 0.101 . | 0.101 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,407 |
Problem 2. The probability of hitting the ring with a ball in one throw is 0.7. Find the probability that in five throws there will be hits in the ring: a) no more than three; b) no less than four. | Solution. Given: $n=5 ; p=0.7 ; q=1-p=0.3$. We need to calculate $p_{5}(k \leqslant 3)$ and $p_{5}(k \geqslant 4)$.
First, let's find $p_{5}(k \geqslant 4)$. We have
$$
\begin{aligned}
& p_{5}(k \geqslant 4)=p_{5}(4)+p_{5}(5)=C_{5}^{4} \cdot 0.7^{4} \cdot 0.3+C_{5}^{5} \cdot 0.7^{5} \cdot 0.3^{0}= \\
&=5 \cdot 0.2401... | 0.5282 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,408 |
Problem 3. Find the most probable number of hits in the ring in five throws, if the probability of hitting the ring with the ball in one throw is $p=0.6$. | Solution. We have $n=5 ; p=0.6 ; q=0.4$. For the number $k_{0}$, we obtain the estimate: $5 \cdot 0.6-0.4 \leqslant k_{0} \leqslant 5 \cdot 0.6+0.6$, i.e., $2.6 \leqslant k_{0} \leqslant 3.6$. Since $k_{0}$ is an integer, then $k_{0}=3$.
Direct calculations of $p_{5}(k)$ lead to the values: $p_{5}(0)=$ $=0.01024 ; p_{... | 3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,409 |
Problem 4. The germination rate of seeds is $90 \%$. What is the probability that out of 7 sown seeds, 5 will germinate? | Solution. The probability of germination of a single seed $p=0.9$, therefore $q=1-p=0.1$ and by the Bernoulli formula (19) we have $p_{7}(5)=$ $=C_{7}^{5} \cdot p^{5} \cdot q^{2}=0.124$.
Answer. 0.124. | 0.124 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,410 |
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