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Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} V_{t t}^{\prime \prime}=9 V_{x x}^{\prime \prime}, \quad V(x, 0)=\left\{\begin{array}{l} \frac{3 x}{40}, \quad 0 \leq x \leq 4 \\ \frac{3(8-x)}{40}, \quad 4 \leq x \leq 8 \end{array},\right. \\ V(0, t)=V(8, t)=0, \quad t \in(0,+\infty) . \end{array} \]

11.5.2.22 Fourier method

7.63 $

Problem: Solve the problem of string oscillations, fixed at the ends, by the method of separation of variables \( x \) (the Fourier method). \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=16 \frac{\partial^{2} u}{\partial x^{2}}, \quad u(0, t)=u(3, t)=0 \\ u(x, 0)=\frac{8}{9}\left(3 x-x^{2}\right),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0 \end{array} \]

11.5.2.23 Fourier method

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Problem: Solve the problem of string oscillations, fixed at the ends, by the method of separation of variables (the Fourier method). \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=25 \frac{\partial^{2} u}{\partial x^{2}}, \quad u(0, t)=u(8, t)=0, \\ u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=\left\{\begin{array}{ll} x, & 0 \leq x<4 \\ 8-x, & 4 \leq x \leq 8 \end{array}\right. \end{array} \]

11.5.2.24 Fourier method

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Problem: 1) Find the Radon-Nikodym density with respect to the Lebesgue measure for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) 2) Find the Jordan expansion for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) What sets can be taken as \( X^{+}, X^{-} \)in Hahn decomposition??

19.6.1.2 Lebesgue measure and integration

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Problem: The random function has the form \( X(t)=V t^{2}(t>0) \), where \( V \) is a random variable, evenly distributed on \( [0 ; 3] \). Find the onedimensional distribution function and onedimensional density of this process.

15.1.28 Theory of random processes

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Problem: Let the random function \( X(t) \) have the characteristics \( \quad m_{x}(t)=1, K_{x}(t, s)=e^{-(t-s)^{2}} \). Find the characteristics of the random functions \[ \begin{array}{l} Y=1+t-x, Z=t^{2}+x^{\prime} \sin t, U=t+x^{\prime \prime}, \\ V=\int_{0}^{t}(1+t) X(t) d t . \end{array} \] Find out wheter the functions \( X, Y, Z, U, V \) are stationary.

15.1.29 Theory of random processes

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Problem: The spectral density of the stationary random function \( X(t) \) is given: \[ S_{X}(\omega)=\frac{1}{\alpha \sqrt{\pi}} \cdot e^{-\frac{\omega^{2}}{4 \alpha^{2}}} . \] Find the correlation function of \( X(t) \).

15.1.30 Theory of random processes

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Problem: Find one-dimensional density, expected value and the dispersion of a random harmonic oscillation \( X(t)=\alpha \cos (\omega t+\phi) \), where \( \alpha, \omega \) are constant, \( \phi \) is a random phase, distributed evenly on \( [-\pi ; \pi] \).

15.1.31 Theory of random processes

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Problem: The correlation function \( K_{X}(\tau) \) of the stationary random function \( X(t) \) is given: \( K_{X}(\tau)=\sigma^{2} e^{-\alpha^{2} \tau^{2}} \). Find the correlation function of the random function \( Y(t)=a X^{\prime}(t) \).

15.1.32 Theory of random processes

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Problem: The random function \( X(t)=t+X_{1} \cos t+ \) \( +X_{2} \sin t \) is given, where the random vector \( \left(\mathrm{X}_{1}, \mathrm{X}_{2}\right) \) has the expected value \( (-1 / 2 ; 1) \) and the correlation matrix \( \left(\begin{array}{cc}3 & -2 \\ -2 & 2,9\end{array}\right) \). Construct a canonical expansion of the process, find its expected value, dispersion and correlation function:

15.1.33 Theory of random processes

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Problem: The spectral density of the stationary random function is given in the form: \( S_{X}(\omega)=\frac{D X}{\pi\left(\alpha^{2}+\omega^{2}\right)} \), where \( D X- \) dispersion of \( X \). Find the correlation function \( K_{X}(\tau) \).

15.1.34 Theory of random processes

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Problem: Find the expected value \( m_{Y}(t) \) and the spectral density \( S_{Y}(\omega) \) of the solutions of the equation \( Y^{\prime \prime}(t)+2 Y^{\prime}(t)+2 Y(t)=X^{\prime}(t)+X(t), \quad \) where \( X(t) \) is the stationary function, and \( m_{X}(t)=1 \), \( K_{X}(\tau)=e^{-|\tau|}(1+|\tau|) \).

15.1.35 Theory of random processes

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Problem: Find the Laplace transform of the function: \[ f(t)=\left\{\begin{array}{ll} 0, & t<0 \\ -2, & 0 \leq t<2 \\ 0, & 2 \leq t<3 \\ \frac{1}{2}, & 3 \leq t<5 \\ 0, & 5 \geq t \end{array}\right. \]

10.4.2 Laplace transform

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Problem: Find the Laplace transform of the function: \[ F(P)=\frac{2 P}{\left(P^{2}+4 P+8\right)^{2}} . \]

10.4.3 Laplace transform

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Problem: Solve the differential equation, using the operational method: \[ y^{\prime \prime}(t)+5 y(t)=e^{3 t}, \quad y(0)=y^{\prime}(0)=0 . \]

8.2.1.5 Differential equations

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