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Example 2. Find the general solution of the linear equation $x y^{\prime}-y=x^{3}$. | Solution. Let $y=u v$, then $y^{\prime}=u^{\prime} v+u v^{\prime}$. Rewrite the equation as
$$
x\left(u^{\prime} v+u v^{\prime}\right)-u v=x^{3} \text { or } u\left(x v^{\prime}-v\right)+x u^{\prime} v=x^{3}
$$
Choose $v$ such that the expression in parentheses becomes zero, then
$$
x \frac{d v}{d x}-v=0, \frac{d v}... | (\frac{x^{2}}{2}+C)x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,822 |
Example 3. Integrate the linear equation $y^{\prime}+y=x+2$. | Solution. Let $y=u v$, then
$$
y^{\prime}=u^{\prime} v+u v^{\prime}
$$
Substituting the expressions for $y$ and $y^{\prime}$ into the equation, we get
$$
u^{\prime} v+u v^{\prime}+u v=x+2, u\left(v^{\prime}+v\right)=x+2-u^{\prime} v
$$
Require that $v^{\prime}+v=0$, then
$$
\frac{d v}{d x}+v=0, \frac{d v}{v}=-d x,... | Ce^{-x}+x+1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,823 |
Example 4. Find the solution of the equation
$$
(x+y) y^{\prime}=1
$$
satisfying the initial condition: $y=0$ when $x=-1$. | Solution. By appearance, this equation is not linear, as it contains the product of the sought function $y$ and its derivative $y^{\prime}$. However, if we consider $x$ as a function of $y$, then, taking into account that
$$
y^{\prime}=\frac{1}{x^{\prime}}
$$
we obtain the linear equation
$$
x^{\prime}=x+y
$$
Let's... | -(x+1) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,824 |
Example 5. Integrate the linear equation
$$
y^{\prime}-y \operatorname{th} x=\frac{1}{2}
$$ | Solution. We apply the method of variation of arbitrary constant. First, we solve the linear homogeneous equation
$$
\begin{gathered}
y^{\prime}-y \operatorname{th} x=0 \\
\frac{d y}{y}=\operatorname{th} x d x, \ln y=\int \operatorname{th} x d x=\ln \operatorname{ch} x+\ln C \\
y=C \operatorname{ch} x
\end{gathered}
$... | \operatorname{ch}x[\operatorname{arctg}e^{x}+C] | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,825 |
Example 6. Integrate the equation
$$
x^{2}(x-1) y^{\prime}-y^{2}-x(x-2) y=0
$$ | Solution. The given equation can be brought to the form
$$
y^{\prime}-\frac{x(x-2)}{x^{2}(x-1)} y=\frac{1}{x^{2}(x-1)} y^{2}
$$
This is a Bernoulli equation, i.e., an equation of the form (9.28). Here,
$$
p(x)=-\frac{x(x-2)}{x^{2}(x-1)} ; q(x)=\frac{1}{x^{2}(x-1)} ; \alpha=2
$$
The substitution (9.29) in this case ... | \frac{x^{2}}{1+C(x-1)} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,826 |
Example 1. Integrate the equation
$$
2 x y d x+x^{2} d y=0
$$ | Solution. The left-hand side of the given equation is the total differential of the function $u(x, y)=x^{2} y$, i.e.,
$$
2 x y d x+x^{2} d y=d\left(x^{2} y\right)
$$
The original equation can be rewritten as
$$
d\left(x^{2} y\right)=0
$$
from which we obtain its general integral
$$
x^{2} y=C
$$ | x^{2}C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,827 |
Example 3. Integrate the equation
$$
\left(y^{3}-2 x y\right) d x+\left(3 x y^{2}-x^{2}\right) d y=0
$$ | Solution. This is an equation of the form (9.30), for which
$$
P(x, y)=y^{3}-2 x y, Q(x, y)=3 x y^{2}-x^{2}
$$
Finding the corresponding partial derivatives, we get:
$$
\frac{\partial P(x, y)}{\partial y}=3 y^{2}-2 x ; \frac{\partial Q(x, y)}{\partial x}=3 y^{2}-2 x
$$
from which
$$
\frac{\partial P}{\partial y}=\... | y^{3}x-x^{2}C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,829 |
Example 4. Integrate the equation
$$
x(x+2 y) d x+\left(x^{2}-y^{2}\right) d y=0
$$ | Solution. In this case
$$
P(x, y)=x(x+2 y), Q(x, y)=x^{2}-y^{2}
$$
from which
$$
\frac{\partial P}{\partial y}=2 x, \frac{\partial Q}{\partial x}=2 x
$$
Condition (9.34) is satisfied, so the given equation is an equation in total differentials. Therefore,
$$
\frac{\partial u}{\partial x}=x^{2}+2 y x, \frac{\partia... | x^{3}+3x^{2}y-y^{3}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,830 |
Example 5. Integrate the equation
$$
\left(x y^{2}-y^{3}\right) d x+\left(1-x y^{2}\right) d y=0 .
$$ | Solution. This equation is not an equation of total differentials, since
$$
\frac{\partial P}{\partial y}=\frac{\partial}{\partial y}\left(x y^{2}-y^{3}\right)=2 x y-3 y^{2}
$$
$$
\frac{\partial Q}{\partial x}=\frac{\partial}{\partial x}\left(1-x y^{2}\right)=-y^{2} \text { and } \frac{\partial P}{\partial y} \neq \f... | x^{2}y-2xy^{2}-2-2C0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,831 |
Example 1. Find the curve passing through the point $M_{0}(1,4)$ and having the property that the segment of any of its tangents, enclosed between the coordinate axes, is bisected at the point of tangency. | Solution. Let $M(x, y)$ be an arbitrary point on the desired curve, and $AB$ be the segment of the tangent to the curve at this point, enclosed between the coordinate axes. By the condition of the problem, $BM = MA$. If $OP$ is the abscissa of point $M$, then $\triangle AMP \sim \triangle ABO$ and
$$
\begin{gathered}
... | xy=4 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,832 |
Example 2. A point moves along a straight line with constant acceleration. Find the law of motion of the point. | Solution. Let's choose time as the independent variable, then the distance traveled and the speed of the point will be functions of time. Denote the distance traveled in time $t$ by $s$, the speed of the point by $v$, and the acceleration by $w$. According to the condition, $w=a$, where $a$ is a constant. Let the initi... | \frac{1}{2}^{2}+v_{0}+s_{0} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,833 |
Example 3. Determine the shape of the mirror reflecting the rays from a point source into a parallel beam. | Solution. Let the source of light be at the origin, and the axis $O x$ be parallel to the reflected rays.
Let $M(x, y)$ be an arbitrary point on the curve, which is the section of the mirror by the plane $O x y$. Draw the normal to the curve at point $M$ (Fig. 9.2) until it intersects the axis $O x$ at point $N(x, 0)$... | y^{2}=2Cx+C^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 32,834 |
Example 4. A vessel containing $10 \pi$ of water is continuously supplied with a solution at a rate of $2 \pi$ per minute, with each liter of the solution containing 0.3 kg of salt. The incoming solution is mixed with the water, and the mixture flows out of the vessel at the same rate. How much salt will be in the vess... | Solution. Let the independent variable be time $t$, and the function we are looking for be $y(t)$ - the amount of salt in the container $t$ minutes after the start of the experiment. We will determine how the amount of salt changes over the time interval from moment $t$ to moment $t+\Delta t$. Into the container, $2 \p... | 1.9 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,835 |
Example 7. In a certain chemical reaction, substance $B$ decomposes into two substances $X$ and $Y$, and the rate of formation of each is proportional to the current amount $b$ of substance $B$. Find the laws governing the changes in the amounts $x$ and $y$ of substances $X$ and $Y$ as functions of time $t$, if at the ... | Solution. If $x$ and $y$ denote the quantities of substances $X$ and $Y$ respectively, then the derivative $\frac{d x}{d t}$ represents the rate of formation of substance $X$, and the derivative $\frac{d y}{d t}$ represents the rate of formation of substance $Y$. At any moment of time $t$, the quantity $b$ of decompose... | \frac{b_{0}}{4}(1-\frac{1}{2^{\}});\frac{3b_{0}}{4}(1-\frac{1}{2^{\}}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,837 |
Example 1. Integrate the equation $y^{\prime \prime}=\cos x$. | Solution. Since $y^{\prime \prime}=\frac{d y^{\prime}}{d x}$, then $\frac{d y^{\prime}}{d x}=\cos x, d y^{\prime}=\cos x d x$, from which
$$
y^{\prime}=\int \cos x d x=\sin x+C_{1}, \frac{d y}{d x}=\sin x+C_{1}
$$
Integrating once more, we get:
$$
\begin{gathered}
y=\int\left(\sin x+C_{1}\right) d x=-\cos x+C_{1} x+... | -\cosx+C_{1}x+C_{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,838 |
Example 2. Find the integral curve of the equation $y^{\prime \prime}=x+1$, passing through the point $M_{0}(1,1)$ and tangent to the line $y=\frac{1}{2} x+\frac{1}{2}$ at this point. | Solution. First, we find the general solution of the given equation. We have:
$$
\begin{gathered}
y^{\prime \prime}=\frac{d y^{\prime}}{d x}, \frac{d y^{\prime}}{d x}=x+1, d y^{\prime}=(x+1) d x, y^{\prime}=\int(x+1) d x \\
y^{\prime}=\frac{x^{2}}{2}+x+C_{1} ; \frac{d y}{d x}=\frac{x^{2}}{2}+x+C_{1}, d y=\left(\frac{x... | \frac{x^{3}}{6}+\frac{x^{2}}{2}-x+\frac{4}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,839 |
Example 3. Find the general solution of the equation $y^{\prime \prime}=2 y^{\prime}$. | Solution. The right-hand side of the given equation depends only on $y^{\prime}$. This is an equation of the form (10.9). Setting $y^{\prime}=p$, we find:
$$
\begin{gathered}
y^{\prime \prime}=\frac{d y^{\prime}}{d x}=\frac{d p}{d x}, \frac{d p}{d x}=2 y^{\prime}=2 p, \frac{d p}{p}=2 d x, \ln p= \\
=2 x+\ln C^{\prime}... | C_{1}e^{2x}+C_{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,840 |
Example 4. Integrate the equation $3 y^{\prime \prime}=y^{-\frac{5}{3}}$. | Solution. This is an equation of the form (10.8), since the right-hand side depends only on $y$. Setting $y^{\prime}=p$, we get $y^{\prime \prime}=p \frac{d p}{d y}$. We rewrite the equation as
$$
3 p \frac{d p}{d y}=y^{-\frac{5}{3}}
$$
Integrating the last equation, we find:
$$
p^{2}=C_{1}-y^{-\frac{2}{3}} \text { ... | x+C_{2}=\\frac{1}{C_{1}^{2}}\sqrt{C_{1}y^{\frac{2}{3}}-1}(C_{1}y^{\frac{2}{3}}+2) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,841 |
Example 5. Integrate the equation
$$
\left(1+x^{2}\right) y^{\prime \prime}-2 x y^{\prime}=0
$$ | Solution. This is an equation of the form (10.11), since the desired function $y$ does not explicitly appear in it. Let $y^{\prime}=p$, then
$$
y^{\prime \prime}=\frac{d y^{\prime}}{d x}=\frac{d p}{d x}
$$
and the equation takes the form
$$
\left(1+x^{2}\right) \frac{d p}{d x}-2 x p=0 \text { or }\left(1+x^{2}\right... | C_{1}(\frac{x^{3}}{3}+x)+C_{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,842 |
Example 6. Integrate the equation $y y^{\prime \prime}-y^{\prime 2}=0$. | Solution. This is an equation of the form (10.12), as it does not explicitly contain the argument $x$. Let $y^{\prime}=p$, then
$$
y^{\prime \prime}=\frac{d y^{\prime}}{d x}=\frac{d y^{\prime}}{d y} \cdot \frac{d y}{d x}=\frac{d p}{d y} \cdot p, y^{\prime \prime}=\frac{d p}{d y} \cdot p
$$
Substituting the expression... | C_{2}e^{C_{1}x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,843 |
Example 1. Find the general solution of the equation $y^{\prime \prime}-5 y^{\prime}+6 y=0$ and identify the particular solution that satisfies the initial conditions: $y=1, y^{\prime}=2$ when $x=0$. | Solution. The characteristic equation (10.16) for the given equation takes the form $k^{2}-5 k+6=0$.
Since this equation has distinct real roots $k_{1}=2, k_{2}=3$, the general solution of the differential equation, according to formula (10.17), is:
$$
y=C_{1} e^{2 x}+C_{2} e^{3 x}
$$
To find the particular solution... | e^{2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,844 |
Example 2. Integrate the equation $y^{\prime \prime}+4 y^{\prime}+4 y=0$. | Solution. The characteristic equation $k^{2}+4 k+4=0$ has equal roots $k_{1}=k_{2}=-2$.
According to formula (10.18), we obtain the general solution of the original differential equation
$$
y=e^{-2 x}\left(C_{1}+C_{2} x\right)
$$ | e^{-2x}(C_{1}+C_{2}x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,845 |
Example 4. Find the general solution of the equation $y^{\prime \prime}+3 y^{\prime}=0$. | Solution. The characteristic equation $k^{2}+3 k=0$ has roots $k_{1}=0, k_{2}=-3$.
The general solution is given by the formula
$$
y=C_{1}+C_{2} e^{-3 x}
$$ | C_{1}+C_{2}e^{-3x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,846 |
Example 5. Integrate the equation $y^{\prime \prime}+4 y=0$. | Solution. The characteristic equation $k^{2}+4=0$ has imaginary roots $k_{1}=2 i, k_{2}=-2 i$.
The general solution of this equation is:
$$
y=C_{1} \cos 2 x+C_{2} \sin 2 x
$$
## Problems
Find the general solutions of homogeneous second-order differential equations with constant coefficients:
1. $y^{\prime \prime}-... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,847 |
Example 2. Find the general solution of the equation
$$
y^{\prime \prime}-y^{\prime}-2 y=e^{2 x}
$$ | Solution. The characteristic equation
$$
k^{2}-k-2=0
$$
has roots $k_{1}=2, k_{2}=-1$, so the general solution of the homogeneous equation
$$
y^{\prime \prime}-y^{\prime}-2 y=0
$$
is given by the formula $y_{0}=C_{1} e^{2 x}+C_{2} e^{-x}$.
The function $f(x)=e^{2 x}$ is obtained from formula (10.23) when $a=1$, $m... | C_{1}e^{2x}+C_{2}e^{-x}+\frac{1}{3}xe^{2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,848 |
Example 3. Integrate the equation
$$
y^{\prime \prime}-4 y^{\prime}+4 y=e^{2 x}
$$ | Solution. The corresponding homogeneous equation
$$
y^{\prime \prime}-4 y^{\prime}+4 y=0
$$
has the general solution
$$
y_{0}=\left(C_{1}+C_{2} x\right) e^{2 x}
$$
since the characteristic equation
$$
k^{2}-4 k+4=0
$$
has equal roots $k_{1}=k_{2}=2$.
A particular solution of the original equation is sought in th... | (C_{1}+C_{2}x)e^{2x}+\frac{1}{2}x^{2}e^{2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,849 |
Example 4. Find the general solution of the equation
$$
y^{\prime \prime}+3 y^{\prime}+2 y=2 x^{2}-4 x-17
$$ | Solution. The right-hand side of the given equation is a second-degree polynomial $f(x)=P_{2}(x)=a x^{2}+b x+c$, where $a=2, b=-4, c=-17$. Since $q \neq 0$, according to formula (10.32), we seek a particular solution in the form
$$
\bar{y}=Q_{2}(x)=A x^{2}+B x+C
$$
Substituting the expressions for $\bar{y}, \bar{y}^{... | C_{1}e^{-x}+C_{2}e^{-2x}+x^{2}-5x-2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,850 |
Example 6. Find the general solution of the equation
$$
y^{\prime \prime}-3 y^{\prime}=x^{2}
$$ | Solution. Since in this case $q=0$ (the term containing $y$ is absent in the left part), according to formula (10.34), we look for a particular solution in the form
$$
\bar{y}=x\left(A x^{2}+B x+C\right), \bar{y}=A x^{3}+B x^{2}+C x
$$
Substituting the expressions
$$
\bar{y}^{\prime}=3 A x^{2}+2 B x+C, \bar{y}^{\pri... | C_{1}+C_{2}e^{3x}-\frac{1}{9}x^{3}-\frac{1}{9}x^{2}-\frac{2}{27}x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,851 |
Example 7. Integrate the equation
$$
y^{\prime \prime}+4 y^{\prime}+4 y=2 \sin 2 x+3 \cos 2 x
$$ | Solution. The right-hand side of the given equation is a trigonometric polynomial of the form $f(x)=a \cos m x+b \sin m x$, where $a=3, b=2$, $m=2$. Since condition (10.28) is satisfied, according to formula (10.27), we seek a particular solution in the form
$$
\bar{y}=A \cos 2 x+B \sin 2 x
$$
Finding the derivatives... | (C_{1}+C_{2}x)e^{-2x}-\frac{1}{4}\cos2x+\frac{3}{8}\sin2x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,852 |
Example 8. Find the general solution of the equation
$$
y^{\prime \prime}-12 y^{\prime}+36 y=\sin 3 x
$$ | Solution. The right-hand side of the given equation is a trigonometric polynomial of the form $f(x)=a \cos m x+b \sin m x$, where $a=0, b=1$, $m=3$. Since condition (10.28) is satisfied, we seek a particular solution in the form
$$
\bar{y}=A \cos 3 x+B \sin 3 x
$$
We find the derivatives
$$
\bar{y}^{\prime}=-3 A \si... | (C_{1}+C_{2}x)e^{6x}+\frac{4}{225}\cos3x+\frac{1}{75}\sin3x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,853 |
Example 2. Solve the equation $x y^{\mathrm{IV}}+y^{\prime \prime \prime}=0$. | Solution. This is an equation of the form (11.2), for which $k=3, n=4$. By taking the lowest derivative $y^{\prime \prime \prime}$ as a new unknown function $u$, we perform the substitution (11.3)
$$
y^{\prime \prime \prime}=u
$$
from which $y^{\mathrm{IV}}=\frac{d u}{d x}$. Thus, the given equation reduces to a firs... | C_{1}^{\}x^{2}\lnx+C_{2}^{\}x^{2}+C_{3}x+C_{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,854 |
Example 3. Solve the equation $y^{\mathrm{IV}}=2 y^{\prime \prime \prime}$. | Solution. Let $y^{\prime \prime \prime}=u$, then $y^{\mathrm{IV}}=u^{\prime}$, so the equation becomes
$$
u^{\prime}=2 u, \frac{d u}{d x}=2 u
$$
Separating variables (assuming $u \neq 0$) and integrating, we get
$$
\ln u=2 x+C, u=e^{2 x+C}=C_{1} e^{2 x}
$$
Since
$$
u=y^{\prime \prime \prime}, \text { then } y^{\pr... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,855 |
Example 2. Solve the equation $y^{\prime \prime \prime}-7 y^{\prime \prime}+15 y^{\prime}-9 y=0$. | Solution. Let's form the characteristic equation
$$
r^{3}-7 r^{2}+15 r-9=0
$$
one of the roots of which $r_{1}=1$ can be obtained by trial and error.
Since
$$
\left(r^{3}-7 r^{2}+15 r-9\right):(r-1)=r^{2}-6 r+9
$$
the equation becomes
$$
(r-1)\left(r^{2}-6 r+9\right)=0
$$
from which $r_{2}=r_{3}=3$. Thus, the ch... | C_{1}e^{x}+(C_{2}+C_{3}x)e^{3x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,856 |
Example 3. Solve the equation $y^{\text {IV }}-16 y=0$. | Solution. The characteristic equation
$$
r^{4}-16=0 \text { or }\left(r^{2}-4\right)\left(r^{2}+4\right)=0
$$
has roots
$$
r_{1}=2, r_{2}=-2, r_{3}=2 i, r_{4}=-2 i
$$
The imaginary conjugate roots $r_{3}$ and $r_{4}$ (for which $\alpha=0, \beta=2$) correspond to the particular solutions
$$
y_{3}=\cos 2 x, y_{4}=\s... | C_{1}e^{2x}+C_{2}e^{-2x}+C_{3}\cos2x+C_{4}\sin2x | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,857 |
Example 4. Solve the equation $y^{\text {IV }}-4 y^{\prime \prime \prime}+8 y^{\prime \prime}-8 y^{\prime}+4 y=0$. | Solution. Let's form the characteristic equation
$$
r^{4}-4 r^{3}+8 r^{2}-8 r+4=0
$$
Transforming the left side of this equation, we get
$$
\begin{gathered}
r^{4}-4 r^{3}+8 r^{2}-8 r+4=r^{4}-4 r^{3}+4 r^{2}+4 r^{2}-8 r+4= \\
=\left(r^{2}-2 r+2\right)^{2}=\left(r^{2}-2 r+2\right)\left(r^{2}-2 r+2\right)=0
\end{gather... | e^{x}[(C_{1}+C_{3}x)\cosx+(C_{2}+C_{4}x)\sinx] | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,858 |
Example 2. Solve the equation $y^{\prime \prime \prime}+y^{\prime}=x^{4}$. | Solution. The right-hand side of the equation is a function of the form (11.18), for which $\alpha=0, m=4$.
The characteristic equation
$$
r^{3}+r=0 \text { or } r\left(r^{2}+1\right)=0
$$
has roots $r_{1}=0, r_{2}=i, r_{3}=-i$, so the general solution of the homogeneous equation $y^{\prime \prime \prime}+y^{\prime}... | C_{1}+C_{2}\cosx+C_{3}\sinx+\frac{x^{5}}{5}-4x^{3}+24x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,859 |
Example 4. Solve the equation
$$
y^{\mathrm{IV}}-y=e^{\alpha x}+e^{-\alpha x}+\cos \beta x
$$ | Solution. The characteristic equation
$$
r^{4}-1=0
$$
has roots $r_{1}=1, r_{2}=-1, r_{3}=i, r_{4}=-i$, therefore
$$
y_{0}=C_{1} e^{x}+C_{2} e^{-x}+C_{3} \cos x+C_{4} \sin x
$$
In accordance with formula (11.25), we seek a particular solution:
$$
\bar{y}=\bar{y}_{1}+\bar{y}_{2}+\bar{y}_{3},
$$
where
$$
\bar{y}_{... | C_{1}e^{x}+C_{2}e^{-x}+C_{3}\sinx+C_{4}\cosx+\frac{e^{\alphax}+e^{-\alphax}}{\alpha^{4}-1}+\frac{\cos\betax}{\beta^{4}-1} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,860 |
Example 5. Solve the equation $y^{\prime \prime \prime}+y^{\prime}=\operatorname{tg} x$. | Solution. The characteristic equation $r^{3}+r=0$ has roots $r_{1}=0, r_{2}=i, r_{3}=-i$, therefore
$$
y_{1}=1, y_{2}=\cos x, y_{3}=\sin x
$$
and the general solution of the homogeneous equation is expressed by the formula
$$
y_{0}=C_{1}+C_{2} \cos x+C_{3} \sin x
$$
Using the method of variation of arbitrary consta... | -\ln\cosx-\sinx\cdot\ln\operatorname{tg}(\frac{x}{2}+\frac{\pi}{4})+C_{2}\cosx+C_{3}\sinx+\bar{C}_{1} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,861 |
Example 1. Solve the system of equations
$$
\frac{d y}{d x}=z, \frac{d z}{d x}=-y
$$ | Solution. This system of two equations with constant coefficients involves two unknown functions $y$ and $z$. Differentiating the first equation with respect to $x$, we find
$$
\frac{d^{2} y}{d x^{2}}=\frac{d z}{d x}
$$
Taking into account the second equation of the system, we obtain
$$
\frac{d^{2} y}{d x^{2}}=-y, \... | C_{1}\cosx+C_{2}\sinx;-C_{1}\sinx+C_{2}\cosx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,862 |
Example 3. Solve the system of equations
$$
\left.\begin{array}{l}
\dot{x}=5 x-3 y+2 e^{3 t} \\
\dot{y}=x+y+5 e^{-t}
\end{array}\right\}
$$
$\mathbf{P e \boldsymbol { e }} \mathbf{\text { e and e }}$. This is a system of the form (11.30), where the dot denotes the derivative with respect to the independent variable $... | Solution. Differentiating the first equation with respect to $t$ and taking into account all the equations of the system, we find
$$
\begin{gathered}
\ddot{x}=\dot{x}+\dot{z}-\dot{y}=(x+z-y)+(2 x-y)-(x+y-z) \\
\ddot{x}=2 x+2 z-3 y
\end{gathered}
$$
Differentiating again, we get
$$
\begin{gathered}
\dddot{x}=2 \dot{x... | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,863 | |
Example 1. Using the method of tangents, find the real root of the equation $x^{3}+x-3=0$. | Solution. Writing the given equation in the form $x^{3}=-x+3$ and plotting the graphs of the functions $f_{1}(x)=x^{3}, f_{2}(x)=-x+3$, we find that the only root of the equation lies in the interval $[1,2]$. Let's specify an interval of lesser length in which the root lies. Since $f(x)=x^{3}+x-3$, $f(1.2)=(1.2)^{3}+1.... | 1.213412 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,866 |
Example 2. Approximate the larger negative root of the equation $x^{3}-12 x-8=0$ using the tangent method with an accuracy of five decimal places after the decimal point. | Solution. Graphically separating the roots of the given equation, we conclude that the equation has three real roots; the larger negative root belongs to the interval $[-1,0]$. We specify an interval of lesser length, in which the root is located; this is the interval $[-0.7; -0.65]$.
Since
$$
f^{\prime \prime}(x)=6 ... | -0.694593 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,867 |
Example 1. Using the method of iterations, find the smaller positive root of the equation $x^{3}-5 x+1=0$. | Solution. Graphically separating the roots of the given equation, we conclude that the equation has three real roots lying in the intervals $[-3 ;-2],[0 ; 1],[2 ; 3]$. Let's find the smallest positive root belonging to the interval $[0 ; 1]$. Specify a segment of lesser length on which the root lies. Since $f(x)=x^{3}-... | 0.20164 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,868 |
Example 3. Find the real roots of the equation $x^{5}+x-3=0$ by the method of iterations. | Solution. Graphically separating the roots of the given equation, we conclude that the equation has a single real root in the interval $[1,2]$. To solve this equation using the method of iterations, it makes no sense to represent it in the form $x=3-x^{5}$, since the function $\varphi(x)=3-x^{5}$ has a derivative $\var... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,869 |
Example 1. Using the formulas for rectangles, taking $n=10$, calculate $J=\int_{0}^{1} \frac{d x}{1+x^{3}}$. | Solution. Here $h=\frac{1-0}{10}=0.1, f(x)=\frac{1}{1+x^{3}}$. We will construct a table of the corresponding values of the argument and function necessary for the approximate calculation of the given integral (Table 13.1).
Table 13.1
| $\mathrm{k}$ | $x_{k}$ | $x_{k}^{3}$ | $1+x_{k}^{3}$ | $y_{k}=f\left(x_{k}\right)... | 0.81002 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,870 |
Example 2. Into how many parts should the interval of integration be divided to calculate $\int_{2}^{7} \frac{d x}{\sqrt{x+2}}$ with an accuracy of 0.1? | Solution. The absolute error in calculating a definite integral using the rectangle method is determined by inequality (13.4). In the problem, the condition $\left|R_{n}(f)\right| \leq \varepsilon$ is set, where $\varepsilon=0.1$. The inequality $\left|R_{n}(f)\right| \leq \varepsilon$ will be satisfied if $\frac{(b-a)... | 8 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,871 |
Example 1. Using the trapezoidal rule, calculate $\int_{0}^{5} \frac{d x}{\sqrt{x+4}}$ for $n=5$. | Solution. Using formulas (13.6), we find $h, x_{k}, y_{k}: h=1$, $x_{k}=k$,
$$
y_{k}=\frac{1}{\sqrt{x_{k}+4}}
$$
$(k=0,1,2,3,4,5): \quad x_{0}=0, \quad x_{1}=1$,
$x_{2}=2, \quad x_{3}=3, \quad x_{4}=4, \quad x_{5}=5 ;$
$$
y_{0}=\frac{1}{\sqrt{4}}=\frac{1}{2}, \quad y_{1}=\frac{1}{\sqrt{5}} \approx 0.447, \quad y_{2... | 2.002 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,872 |
Example 2. Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$. | Solution. To determine the number $n$ of segments into which the interval of integration should be divided, we will use inequality (13.7). The inequality $\left|R_{n}\right| \leq \varepsilon$ will be satisfied if $\frac{(b-a)^{3} M}{12 n^{2}} \leq \varepsilon$, from which
$$
n \geq \sqrt{\frac{(b-a)^{3} M}{12 \varepsi... | 0.6956 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,873 |
Example 2. Using the parabola formula with an accuracy of 0.0001, calculate $\int_{0}^{1} \frac{d x}{1+x^{2}}$ | Solution. We will use formula (13.10). The inequality $\left|R_{n}\right| \leq \varepsilon$ will be satisfied when
$$
\frac{(b-a)^{5} \cdot M}{180 \cdot(2 n)^{4}} \leq \varepsilon, \text { i.e., when } 2 n \geq \sqrt[4]{\frac{(b-a)^{5} \cdot M}{180 \varepsilon}}
$$
We find the value $M=\max _{a \leq x \leq b}\left|f^... | 0.785395 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,875 |
Example 1. Using the Euler method, find the values of the solution of the equation $y^{\prime}=y+x^{2}$, satisfying the condition $y(0)=1$, at the first five points of the interval $[0 ; 0.5]$ for $h=0.1$. | Solution. According to formula (14.2), we find the points $x_{0}=0 ; x_{1}=0.1$; $x_{2}=0.2 ; \quad x_{3}=0.3 ; \quad x_{4}=0.4 ; \quad x_{5}=0.5$. The values of the sought function $y=y(x)$, satisfying the conditions of the given Cauchy problem, are found in accordance with formula (14.3). The results of the calculati... | 1.6426 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,876 |
Example 2. Using Euler's method, find the values of the solution of the differential equation $y^{\prime}=2 x-y$, for which $y(1)=1$, at five points on the interval $[1 ; 1.5]$, taking $h=0.1$. | Solution. Based on formula (14.2), we determine the points $x_{0}=1$; $x_{1}=1.1 ; x_{2}=1.2 ; x_{3}=1.3 ; x_{4}=1.4 ; x_{5}=1.5$. The values of the sought function $y=y(x)$, satisfying the condition of the given Cauchy problem, are calculated using formula (14.3). The results of the calculations are recorded in Table ... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,877 |
Example 1. Using the Runge-Kutta method, find the solution to the Cauchy problem for the equation $y^{\prime}=y-x^{2}, y(1)=0, x \in[1,2]$ at the first five points, taking $h=0.1$. | Solution. Since in this case $f(x, y)=y-x^{2}$ and due to the condition $x_{0}=1, y_{0}=0$, then $f\left(x_{0}, y_{0}\right)=y_{0}-x_{0}^{2}=0-1=-1$. Using formulas (14.9), we find: $k_{1}^{0}=h f\left(x_{0}, y_{0}\right)=0.1(-1)=-0.1 ; k_{2}^{0}=0.1 f(1.05, -0.05)=$ $=0.1\left[(-0.05)-(1.05)^{2}\right]=-0.1152$; $k_{3... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,878 |
Example 2. Using the Runge-Kutta method, find the solution to the Cauchy problem for the equation $y^{\prime}=3 x+y, y(0)=-1, x \in[0,1]$, taking $h=0.1$. | Solution. As in the previous example, using formulas (14.7) - (14.9), we find the values of $y_{1}, y_{2}, y_{3}, y_{4}, y_{5}$ (see Table 14.4).
Table 14.4
| i | $x_{i}$ | $y_{i}$ | $3 x_{i}$ | $3 x_{i}+y_{i}$ | $k_{j}$ | $p_{j}$ | $S=$ $=\sum_{j=1}^{4} p_{j} k_{j}$ | $\Delta y_{i}=$ $=\frac{1}{6} s$ |
| :---: | :... | 2e^{x}-3x-3 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,879 |
Example 1. Find the Lagrange interpolation polynomial that takes the values $y_{0}=-5, y_{1}=-11, y_{2}=10$ at the points $x_{0}=-3, x_{1}=-1, x_{2}=2$. | Solution. When $n=2$, formula (15.1) takes the form
$$
P_{2}(x)=y_{0} \frac{\left(x-x_{1}\right)\left(x-x_{2}\right)}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}+y_{1} \frac{\left(x-x_{0}\right)\left(x-x_{2}\right)}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)}+y_{2} \frac{\left(x-x_{0}\right)\left(x-x_{1}\r... | 2x^{2}+5x-8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,880 |
Example 2. Find the Lagrange interpolation polynomial $P(x)$, for which $P(-1)=-11, P(1)=-3, P(2)=1, P(3)=13$. | Solution. In
this
case
$x_{0}=-1, \quad x_{1}=1, \quad x_{2}=2, \quad x_{3}=3, \quad y_{0}=-11, \quad y_{1}=-3, \quad y_{2}=1, \quad y_{3}=13$. For $n=3$, formula (15.1) takes the form
$$
\begin{aligned}
& P(x)=y_{0} \frac{\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right)}{\left(x_{0}-x_{1}\right)\left(x... | P_{3}(x)=x^{3}-2x^{2}+3x-5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,881 |
Example 1. Construct a table of differences of various orders for the following values of $x$ and $f(x)$:
\[
\begin{gathered}
x_{0}=-1, x_{1}=-2, x_{2}=1, x_{3}=2, x_{4}=3 \\
y_{0}=0, y_{1}=7, y_{2}=30, y_{3}=-16, y_{4}=-45
\end{gathered}
\] | Solution. Using formulas (15.4), we find the first differences:
$$
\begin{gathered}
\Delta y_{0}=y_{1}-y_{0}=7-0=7 ; \Delta y_{1}=y_{2}-y_{1}=30-7=23 \\
\Delta y_{2}=y_{3}-y_{2}=-16-30=-46 ; \Delta y_{3}=y_{4}-y_{3}=-45-(-16)=-29
\end{gathered}
$$
In accordance with formulas (15.5), we calculate the second-order diff... | 171 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,882 |
Example 2. Construct a table of divided differences of various orders for the following values of $x$ and $y=f(x)$:
\[
\begin{gathered}
x_{0}=-3, x_{1}=-2, x_{2}=-1, x_{3}=1, x_{4}=2 \\
y_{0}=-9, y_{1}=-16, y_{2}=-3, y_{3}=11, y_{4}=36
\end{gathered}
\] | Solution. According to the definitions, we find the divided differences of the first order:
$$
\begin{gathered}
f\left(x_{1}, x_{0}\right)=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}=\frac{-16-(-9)}{-2-(-3)}=-7 ; f\left(x_{2}, x_{1}\right)=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-3-(-16)}{-1-(-2)}=13 \\
f\left(x_{3}, x_{2}\right)=\... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,883 |
Example 3. Given the polynomial $P(x)=x^{3}-2 x^{2}+3 x+1$. Known are its
$$
x_{0}=0, \quad x_{1}=1, \quad x_{2}=2, \quad x_{3}=3:
$$
$P(0)=1, \quad P(1)=3, \quad P(2)=7, P(3)=19$. Find the values of the polynomial at $x_{k}=k \quad(k=4,5,6,7,8,9,10)$. | Solution. We construct a table of differences of various orders (Table 15.4).
Table 15.4
Table 15.5
| $x$ | $y$ | $\Delta y$ | $\Delta^{2} y$ | $\Delta^{3} y$ |
| :---: | :---: | :---: | :---: | :---: |
| 0 | 1 | 2 | | |
| 1 | 3 | 4 | 2 | 6 |
| 2 | 7 | 12 | 8 | |
| 3 | 19 | | | |
| | | | | |
| $x$ | $y$... | P(5)=91,P(6)=163,P(7)=267,P(8)=409,P(9)=595,P(10)=831 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,884 |
Example 1. Find the Newton interpolation polynomial for the function $y=f(x)$ if its values are known: $f(2)=1, f(4)=15, f(5)=28$. | Solution. In this case
$$
x_{0}=2, x_{1}=4, x_{2}=5
$$
$$
y_{0}=1, y_{1}=15, y_{2}=28
$$
Note that the nodes are not equidistant (since $x_{1}-x_{0} \neq x_{2}-x_{1}$). The interpolating polynomial (15.14) for $n=2$ takes the form
$$
P_{2}(x)=y_{0}+\left(x-x_{0}\right) f\left(x_{1}, x_{0}\right)+\left(x-x_{0}\right... | P_{2}(x)=2x^{2}-5x+3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,885 |
Example 2. Find the Newton polynomials for "forward interpolation" and "backward interpolation" for the function given by the table
| $x$ | 0 | 1 | 2 | 3 |
| :---: | :---: | :---: | :---: | :---: |
| $y$ | 5 | 1 | 7 | 29 |
Calculate the values of the function at $x=0.5$ and $x=2.5$. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,886 | |
Example 1. Two dice are rolled, and the sums of the points that come up (the sums of the numbers on the top faces of both dice) are counted. The sum of the points on the two dice can vary from 2 to 12. Write the complete set of events in this experiment. | Solution. The complete set of events is formed by equally possible elementary outcomes $(k ; m), k, m=1,2,3,4,5,6$, presented in Table 1.1. The elementary outcome ( $k ; m$ ) means that $k$ points fell on the first die, and $m$ points on the second die ( $k, m=1,2,3,4,5,6$ ). For example, $(3 ; 4)$ means 3 points on th... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,887 |
Example 2. How many elementary outcomes favor the event "the same number of points fell on both dice" when two gaming dice are rolled? | Solution. This event is favored by 6 elementary outcomes (see Table 1.1$):(1 ; 1),(2 ; 2),(3 ; 3),(4 ; 4),(5 ; 5),(6 ; 6)$. | 6 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,888 |
Example 4. Three dice are rolled, and the sums of the points that fall on them are counted. In how many ways can a total of 5 points, 6 points be obtained? | Solution. A total of 5 points can be obtained in six ways: $(1 ; 1 ; 3),(1 ; 3 ; 1),(3 ; 1 ; 1),(1 ; 2 ; 2),(2 ; 1 ; 2),(2 ; 2 ; 1)$. A total of 6 points can be obtained in ten ways: $(1 ; 1 ; 4),(1 ; 4 ; 1),(4 ; 1 ; 1),(1 ; 2 ; 3),(1 ; 3 ; 2),(2 ; 1 ; 3)$, $(2 ; 3 ; 1),(3 ; 1 ; 2),(3 ; 2 ; 1),(2 ; 2 ; 2)$
3 remark. T... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,889 |
Example 1. In an urn, there are 10 balls of the same size and weight, of which 4 are red and 6 are blue. One ball is drawn from the urn. What is the probability that the drawn ball will be blue? | Solution. Let the event "the drawn ball is blue" be denoted by the letter $A$. This trial has 10 equally likely elementary outcomes, 6 of which are favorable to event $A$. According to formula (1.2.1), we get
$$
P(A)=\frac{6}{10}=0.6
$$ | 0.6 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,890 |
Example 2. All natural numbers from 1 to 30 are written on identical cards and placed in an urn. After thoroughly mixing the cards, one card is drawn from the urn. What is the probability that the number on the drawn card will be a multiple of $5?$ | Solution. Let $A$ be the event "the number on the taken card is divisible by 5". In this trial, there are 30 equally likely elementary outcomes, of which 6 outcomes are favorable to event $A$ (the numbers $5,10,15,20,25,30$). Therefore,
$$
P(A)=\frac{6}{30}=0.2
$$ | 0.2 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 32,891 |
Example 3. Two dice are rolled, and the sum of the points on the upper faces is calculated. Find the probability of event $B$, which consists of the sum of the points on the upper faces of the dice being 9. | Solution. In this trial, there are $6^{2}=36$ equally likely elementary outcomes (see Table 1.1). Event $B$ is favored by 4 outcomes: $(3 ; 6),(4 ; 5),(5 ; 4),(6 ; 3)$, therefore
$$
P(B)=\frac{4}{36}=\frac{1}{9}
$$ | \frac{1}{9} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,892 |
Example 4. A natural number not exceeding 10 is chosen at random. What is the probability that this number is prime? | Solution. Let $C$ be the event "the selected number is prime". In this case, $n=10, m=4$ (prime numbers $2,3,5,7$). Therefore, the required probability
$$
P(C)=\frac{4}{10}=0.4
$$ | 0.4 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 32,893 |
Example 6. What is the probability that in a randomly chosen two-digit number, the digits are the same? | Solution. Two-digit numbers are the numbers from 10 to 99; there are 90 such numbers in total. Numbers with identical digits are 9 (these are the numbers 11, 22, 33, $44,55,66,77,88,99$). Since in this case $m=9, n=90$, then
$$
P(A)=\frac{9}{90}=0.1,
$$
where $A$ is the event "number with identical digits". | 0.1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,895 |
Example 7. A letter is randomly selected from the word "differential". What is the probability that this letter will be: a) a vowel, b) a consonant, c) the letter $4?$ | Solution. The word differential has 12 letters, of which 5 are vowels and 7 are consonants. The letter ч is not in this word. Let's denote the events: $A$ - "vowel letter", $B$ - "consonant letter", $C$ - "letter ч". The number of favorable elementary outcomes: $m_{1}=5$ - for event $A, m_{2}=7$ - for event $B$, $m_{3}... | P(A)=\frac{5}{12}\approx0.417;\quadP(B)=\frac{7}{12}\approx0.583;\quadP(C)=0 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,896 |
Example 9. The book has 300 pages. What is the probability that a randomly opened page will have an ordinal number that is a multiple of 5? | Solution. From the condition of the problem, it follows that the total number of equally possible elementary outcomes, forming a complete group of events, is $n=300$. Of these, $m=60$ are favorable for the occurrence of the specified event. Indeed, a number divisible by 5 has the form $5k$, where $k$ is a natural numbe... | 0.2 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 32,897 |
Example 10. Two dice are rolled, and the sum of the points on the upper faces is calculated. What is more likely - to get a sum of 7 or 8? | Solution. Let's denote the events: $A$ - "7 points have fallen", $B$ - "8 points have fallen". Event $A$ is favorable in 6 elementary outcomes: $(1 ; 6)$, $(2 ; 5)$, $(3 ; 4)$, $(4 ; 3)$, $(5 ; 2)$, $(6 ; 1)$, and event $B$ in 5 outcomes: $(2 ; 6)$, $(3 ; 5)$, $(4 ; 4)$, $(5 ; 3)$, $(6 ; 2)$. The total number of equall... | P(A)>P(B) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,898 |
Example 1. In how many different ways can three people be selected for three different positions from ten candidates? | Solution. We will use formula (1.3.3). For $n=10, m=3$ we get
$$
A_{10}^{3}=10 \cdot 9 \cdot 8=720
$$ | 720 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,899 |
Example 3. In how many ways can three people be selected for three identical positions from ten candidates? | Solution. According to formula (1.3.4), we find
$$
C_{10}^{3}=\frac{10!}{3!\cdot(10-3)!}=\frac{10!}{3!\cdot 7!}=\frac{10 \cdot 9 \cdot 8}{1 \cdot 2 \cdot 3}=120
$$ | 120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,900 |
Example 4. How many different six-digit numbers can be written using the digits $1 ; 1 ; 1 ; 2 ; 2 ; 2$? | Solution. Here we need to find the number of permutations with repetitions, which is determined by formula (1.3.7). For $k=2, n_{1}=3, n_{2}=3, n=6$ this formula gives us
$$
P_{6}(3 ; 3)=\frac{6!}{3!\cdot 3!}=\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}{1 \cdot 2 \cdot 3 \cdot 1 \cdot 2 \cdot 3}=20
$$ | 20 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,901 |
Example 5. How many different permutations of letters can be made from the words: zamok, rotor, topor, kolokol? | Solution. In the word "замок" (lock), all letters are different, and there are five of them. According to formula (1.3.1), we get
$$
P_{5}=5!=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5=120
$$
In the word "ротор" (rotor), consisting of five letters, the letters $p$ and о are repeated twice. To count the different permutations,... | 210 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,902 |
Example 8. On six cards of the same shape and size, the letters of the word talent are written - one letter on each card. The cards are thoroughly shuffled. They are randomly drawn and placed on the table one after another. What is the probability of getting the word $m a$ lant again? | Solution. Let's number the cards with letters:
| 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathrm{a}$ | $\mathrm{a}$ | $\mathrm{T}$ | $\mathrm{H}$ | $\mathrm{r}$ | $\mathrm{r}$ |
The word "talant" will not change if the letters $a$ are swapped, 513246 but the arrangement of the car... | \frac{1}{180} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,903 |
Example 9. On five identical cards, letters are written: on two cards $l$, on the other three $i$. These cards are randomly laid out in a row. What is the probability that the word "lilii" will be formed? | Solution. Let's find the number of permutations of these five letters with repetitions. According to formula (1.3.7) for $n=5, n_{1}=2, n_{2}=3$, we get
$$
P_{5}(2 ; 3)=\frac{5!}{2!\cdot 3!}=\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5}{1 \cdot 2 \cdot 1 \cdot 2 \cdot 3}=10
$$
This is the total number of equally possible ... | 0.1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,904 |
Example 11. Among 25 students in a group, where 10 are girls, 5 tickets are being raffled. Find the probability that 2 of the ticket winners will be girls. | Solution. The number of all equally possible cases of distributing 5 tickets among 25 students is equal to the number of combinations of 25 elements taken 5 at a time, i.e., $C_{25}^{5}$. The number of groups of three boys out of 15 who can receive tickets is $C_{15}^{3}$. Each such trio can be combined with any pair o... | \frac{195}{506}\approx0.385 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,905 |
Example 12. In a box, there are 15 red, 9 blue, and 6 green balls. Six balls are drawn at random. What is the probability that 1 green, 2 blue, and 3 red balls are drawn (event $A$)? | Solution. There are a total of 30 balls in the box. In this experiment, the number of all equally possible elementary outcomes will be $C_{30}^{6}$. Let's calculate the number of elementary outcomes favorable to event $A$. Three red balls out of 15 can be chosen in $C_{15}^{3}$ ways, two blue balls out of 9 can be chos... | \frac{24}{145}\approx0.17 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,906 |
Example 13. In a box, there are 15 balls, of which 5 are blue and 10 are red. Six balls are randomly selected. Find the probability that among the selected balls, 2 are blue. | Solution. The total number of elementary outcomes of the given experiment is equal to the number of combinations of 15 taken 6 at a time, i.e.
$$
C_{15}^{6}=\frac{15!}{6!\cdot 9!}=\frac{15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}=13 \cdot 11 \cdot 7 \cdot 5=5005
$$
The ... | 0.4196 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,907 |
Example 14. A die is rolled 10 times. What is the probability that the faces $1,2,3,4,5,6$ will appear respectively $2,3,1,1,1,2$ times (event $A$)? | Solution. The number of outcomes favorable to event $A$ is calculated using formula (1.3.7):
$$
m=\frac{(2+3+1+1+1+2)!}{2!\cdot 3!\cdot 1!\cdot 1!\cdot 1!\cdot 2!}=\frac{10!}{4 \cdot 6}=2 \cdot 3 \cdot 5 \cdot 7 \cdot 8 \cdot 9 \cdot 10
$$
The number of all elementary outcomes in this experiment is $n=6^{10}$, theref... | 0.002 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,908 |
Example 1. Out of 500 randomly taken parts, 8 were defective. Find the frequency of defective parts. | Solution. Since in this case $m=8, n=500$, then according to formula (1.4.1) we find
$$
W=\frac{8}{500}=0.016
$$ | 0.016 | Other | math-word-problem | Yes | Yes | olympiads | false | 32,909 |
Example 2. A die was rolled 60 times, during which a six appeared 10 times. What is the frequency of the six appearing? | Solution. From the condition of the problem, it follows that $n=60, m=10$, therefore
$$
W=\frac{10}{60}=\frac{1}{6}
$$ | \frac{1}{6} | Other | math-word-problem | Yes | Yes | olympiads | false | 32,910 |
Example 3. Among 1000 newborns, there were 515 boys. What is the frequency of boys being born? | Solution. Since in this case $n=1000, m=515$, then
$$
W=\frac{515}{1000}=0.515
$$ | 0.515 | Other | math-word-problem | Yes | Yes | olympiads | false | 32,911 |
Example 4. As a result of 20 shots at the target, 15 hits were obtained. What is the frequency of hits? | Solution. Since $n=20, m=15$, then
$$
W=\frac{15}{20}=\frac{3}{4}=0.75
$$ | 0.75 | Other | math-word-problem | Yes | Yes | olympiads | false | 32,912 |
Example 5. When shooting at a target, the frequency of hits $W=0.75$. Find the number of hits in 40 shots. | Solution. From formula (1.4.1), it follows that $m=W n$. Since $W=0.75, n=40$, then $m=0.75 \cdot 40=30$. Therefore, 30 hits were obtained. | 30 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,913 |
Example 6. The frequency of normal seed germination $W=0.97$. Out of the sown seeds, 970 germinated. How many seeds were sown? | Solution. From formula (1.4.1), it follows that $n=\frac{m}{W}$. Since $m=970, \quad W=0.97$, then $n=970 / 0.97=1000$. Therefore, 1000 seeds were sown. | 1000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,914 |
Example 7. In the segment of the natural number series from 1 to 20, find the frequency of prime numbers. | Solution. On the specified segment of the natural number series, the following prime numbers are found: $2,3,5,7,11,13,17,19$; there are 8 in total. Since $n=20, m=8$, the sought frequency is
$$
W=\frac{8}{20}=0.4
$$ | 0.4 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 32,915 |
Example 8. Three series of multiple coin tosses of a symmetric coin were conducted, and the numbers of coat of arms appearances were counted: 1) $n_{1}=4040$, $m_{1}=2048$, 2) $n_{2}=12000, m_{2}=6019$; 3) $n_{3}=24000, m_{3}=12012$. Find the frequency of the coat of arms appearing in each series of trials.
$\mathbf{S ... | Solution. In this case $n=300, m=15$; therefore
$$
W=\frac{15}{300}=0.05
$$ | 0.05 | Other | math-word-problem | Yes | Yes | olympiads | false | 32,916 |
Example 10. A controller, checking the quality of 400 items, found that 20 of them are of second grade, and the rest are of first grade. Find the frequency of first-grade items, and the frequency of second-grade items. | Solution. First, let's find the number of first-grade items: $400-20=380$. Since $n=400, m_{1}=380$, the frequency of first-grade items is
$$
W_{1}=\frac{380}{400}=0.95
$$
Similarly, we find the frequency of second-grade items:
$$
W_{2}=\frac{20}{400}=0.05
$$
## Problems
1. The quality control department found 10 ... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 32,917 |
Example 1. A square is inscribed in a circle (Fig. 1.2). A point is randomly thrown into the circle. What is the probability that the point will land in the square? | Solution. Let's introduce the following notations: $R$ - the radius of the circle, $a$ - the side of the inscribed square, $A$ - the event of a point falling into the square, $S$ - the area of the circle, $S_{1}$ - the area of the inscribed square. As is known, the area of the circle $S=\pi R^{2}$. The side of the insc... | \frac{2}{\pi}\approx0.637 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 32,918 |
Example 2. In a square (Fig. 1.3) with vertices at points $O(0,0)$, $K(0,1)$, $L(1,1)$, $M(1,0)$, a point $Q(x, y)$ is randomly thrown. Find the probability that the coordinates of this point satisfy the inequality $y>\frac{1}{2} x$. | Solution. Draw the line $y=(1 / 2) x$, it will intersect the segment $M L$ at the point $N(1 ; 1 / 2)$. This line divides the plane into two half-planes: for the coordinates of points in the first one (upper) the inequality $y>x / 2$ will be satisfied, for the second (lower) - the inequality $y<x / 2$. The points for w... | 0.75 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 32,919 |
Example 3. (Buffon's Problem). A plane is marked with parallel lines, the distance between which is $a$. A segment of length $l(l<a)$ is thrown randomly onto this plane. What is the probability that the segment intersects at least one of the lines in the family?
. The angle between the segment and the ray parallel to the family of straight lines, whose origin coincides with the upper end of the segment, is denoted by $x$. Clearly, $0 \leq y \leq a$, $0 \l... | \frac{2}{\pi} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,920 |
Example 4. A cube is inscribed in a sphere. A point is randomly fixed inside the sphere. Find the probability that the point will fall inside the cube. | Solution. Let's introduce the following notations: event $A$ - "the point falls into the cube"; $R$ - radius of the sphere, $a$ - edge of the cube, $V$ - volume of the sphere, $V_{1}$ - volume of the inscribed cube.
As is known, $V=\frac{4}{3} \pi R^{3}$; since $V_{1}=a^{3}$ and $a=2 R / \sqrt{3}$, then $V_{1}=\frac{8... | \frac{2}{\pi\sqrt{3}}\approx0.368 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 32,921 |
Example 5. On the plane, the region $G$ is bounded by the ellipse $x^{2} / 49 + y^{2} / 16 = 1$, and the region $g$ - by the ellipse $x^{2} / 25 + y^{2} / 9 = 1$ (Fig. I.6). A point is thrown into the region $G$. What is the probability that the point will land in the region $g?$ | Solution. First, we compute the area of the region bounded by the ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1$, by writing its parametric equations: $x=a \cos t, y=b \sin t$. Since the ellipse is symmetric with respect to the coordinate axes, it is sufficient
.
. Since in this case $a=2 \sqrt{2}, b=\sqrt{2}$, then $S=4 \pi$.
To calculate the area of the region $g$, we will use the formula
$$
S=\... | 0.303 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,923 |
Example 9. Two steamships must approach the same pier. The arrival time of both steamships is independent and uniformly distributed over the given day. Determine the probability that one of the steamships will have to wait for the pier to be freed, if the first steamship's stay time is one hour, and the second's is two... | Solution. Let $x$ and $y$ be the arrival times of the steamships. Possible values for $x$ and $y: 0 \leq x \leq 24, 0 \leq y \leq 24$. Favorable values are $y-x \leq 1, x-y \leq 2$. These inequalities define the area shaded in Fig. 1.13. The area of this region is $S_{g}=24 \cdot 24-0.5 \cdot 23 \cdot 23-0.5 \cdot 22 \... | 0.121 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 32,926 |
Example 10. Two positive numbers, each not greater than one, are taken at random. What is the probability that their sum does not exceed one and their product is no more than $2 / 9?$ | Solution. Let the taken numbers be denoted by $x$ and $y$. Their possible values satisfy the inequalities: $0 \leq x \leq 1, 0 \leq y \leq 1$, which on the plane define a unit square with area $S_{G}=1$. Favorable values of $x$ and $y$ are determined by the conditions: $x+y \leq 1$, $x \cdot y \leq 2 / 9$; the region $... | 0.467 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 32,927 |
Example 11. A point is thrown into a rectangle with vertices $K(-1,0), L(-1,5)$, $M(2,5), N(2,0)$. What is the probability that its coordinates $(x, y)$ will satisfy the inequalities $x^{2}+1 \leq y \leq x+3$? | Solution. The points whose coordinates satisfy the given inequalities belong to the region $g$, bounded by the parabola $y=x^{2}+1$ and the line $y=x+3$ (Fig. 1.15). These lines intersect at points $Q(-1,2)$ and $M(2,5)$. Let's calculate the area of region $g$:
$$
\begin{aligned}
& S_{g}=\int_{-1}^{2}\left(y_{2}-y_{1}... | 0.3 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 32,928 |
Example 12. The region $G$ is bounded by the ellipsoid $\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{4}=1$, and the region $g$ - by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly fixed in the region $G$. What is the probability that it belongs to the region $g$ (event $A$)? | Solution. The volume $V$ of the body bounded by the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ is given by the formula
$$
V=\frac{4}{3} \pi a b c
$$
therefore
$$
V_{\mathrm{G}}=\frac{4}{3} \pi 4 \cdot 3 \cdot 2=32 \pi
$$
The volume of a sphere of radius $R$ is given by the formula
$$... | \frac{2}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 32,929 |
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