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Problem: Find the general solution of the first order inhomogeneous differential equation: \[ y^{\prime} \cos x+y=1-\sin x \text {. } \]

8.1.1.1 First order differential equations

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Problem: Find the general solution of the first order inhomogeneous differential equation: \[ y^{\prime}+\frac{x y}{4+x^{2}}=\frac{1}{\sqrt{4+x^{2}}} \text {. } \]

8.1.1.2 First order differential equations

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Problem: Expand the function \( y(x) \) into a power series in the vicinity of the point \( x_{0} \), where \[ y=\cos ^{2}\left(\frac{\pi x}{6}\right), \quad x_{0}=3 \text {. } \]

2.8.15 Power series

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Problem: Calculate the value of the integral with an accuracy of \( \varepsilon=0,01 \), expanding the integrand function into a power series. \[ \int_{0}^{0,5} \frac{d x}{\sqrt[3]{27+x^{3}}} \]

2.8.17 Power series

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Problem: Find the general solution of the second order inhomogeneous differential equation. \[ 3 y^{\prime \prime}-4 y^{\prime}+2 y=3 e^{2 x} \text {. } \]

8.1.2.3 Second order differential equations

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T Problem: Calculate the area of the domain \( D \), bounded by the given lines: \( D:\left\{\begin{array}{c}y=(x+4) e^{-x} \\ y=0, \quad x=0\end{array}\right. \)

9.7.1 The area of a region

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(a) Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{c} y=\sqrt{18-x^{2}} \\ y=3 \sqrt{2}-\sqrt{18-x^{2}} \end{array}\right. \]

9.7.2 The area of a region

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3 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{c} y=\frac{2}{x}, \quad y=2 \sqrt{x} \\ y=1 \end{array}\right. \]

9.7.3 The area of a region

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Problem: Find the general solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{l} x_{t}^{\prime}=-x+2 y+\frac{3 e^{t}}{t^{2}+4} \\ y_{t}^{\prime}=-2 x+3 y-\frac{2 e^{t}}{t^{2}+4} \end{array}\right. \]

8.3.1 Systems of ordinary differential equations

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Problem: Investigate the stability of the system in at zeroresting point. \[ \left\{\begin{array}{c} \frac{d y}{d t}=-\sin y-x e^{-x} \\ \frac{d x}{d t}=\sin (-x)+\sin (-y) \end{array} .\right. \]

8.4.1.1 Stability of the systems of equations

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3) Problem: Calculate the surface integral of the first kind of the function \( \vec{F} \) over the surface \( S \), where \[ \begin{array}{l} \vec{F}(x, y, z)=\left\{y^{2}-z^{2}, z^{2}-x^{2}, x^{2}-y^{2}\right\}, \\ S:\left\{\begin{array}{c} 0 \leq x \leq 4,0 \leq y \leq 4,0 \leq z \leq 4 \\ x+y+z=6 \end{array}\right. \end{array} \]

9.8.1 Surface integrals

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Problem: The table shows the numbers of sets of arguments (in lexicographic order), on which the logical function takes a value equal to one. Write this function in PDNF, PCNF, PPNF. Minimize it using the Quine method and the Karnaugh method. \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline \multicolumn{1}{|c|}{ Constituent numbers } \\ \hline 2 & 3 & 6 & 7 & 8 & 14 & 15 & - \\ \hline \end{tabular}

6.3.2 Boolean algebra

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3 Problem: Calculate using the Ostrogradsky formula: \[ \int_{S} \int^{(x \cos \alpha+y \cos \beta+z \cos \gamma)} \underset{\sqrt{x^{2}+y^{2}+z^{2}}}{d s} \] where \( S=\left\{x^{2}+y^{2}+z^{2}=z\right\} \), \( \vec{n}=(\cos \alpha, \cos \beta, \cos \gamma) \) is the outside normal.

9.8.2 Surface integrals

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3) Problem: Calculate the volume of the solid of revolution obtained by the rotation of the figure \( D \) around the axis \( O X \), where \[ D:\left\{\begin{array}{c} y=x+\sin x \\ y=0, \quad x=0, \quad x=\pi \end{array}\right. \]

9.6.1 Volume of a solid of revolution

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全 Problem: Calculate the volume of the solid formed by rotating around the axis \( O y \) the figure bounded by the lines: \( \frac{x^{2}}{9}-\frac{y^{2}}{4}=1, x=6 \). Let's plot the drawing.

9.6.2 Volume of a solid of revolution

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