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Problem: Find the general solution of the second order inhomogeneous differential equation \[ y^{\prime \prime}-2 y^{\prime}-3 y=-5 e^{-6 x}-4 e^{3 x} \text {. } \]

8.1.2.6 Second order differential equations

1.28 $

Problem: Integrate the following linear differential equation, the right side of which is presented in the graph, shown in the figure. \[ \begin{array}{l} \text { When } \quad t=0, x=0, x^{\prime}=0, \\ x^{\prime \prime}+x=f(t) . \end{array} \] clip2net.com

8.2.1.2 Differential equations

5.11 $

Problem: Solve the Cauchy problem, using the operational method. \[ 2 y^{\prime \prime}+5 y^{\prime}=29 \cos t, y(0)=-1, \quad y^{\prime}(0)=0 . \] clip2net.com

8.2.1.4 Differential equations

2.55 $

Problem: Solve the inhomogeneous system of differential equations, using the operational method. \[ \left\{\begin{array}{ll} x_{1}^{\prime}=5 x_{1}-18 x_{2}-28 e^{-t} & x_{1}(0)=-1 \\ x_{2}^{\prime}=2 x_{1}-7 x_{2}-9 e^{-t} & x_{2}(0)=-2 \end{array}\right. \]

8.2.2.2 Systems of differential equations

3.83 $

Problem: Solve the inhomogeneous system of differential equations, applying the operational method. \[ \left\{\begin{array}{cc} x_{1}^{\prime}=4 x_{1}-16 x_{2}+11 x_{3}-10 & x_{1}(0)=5 \\ x_{2}^{\prime}=-3 x_{1}+9 x_{2}-7 x_{3}+8, & x_{2}(0)=0 \\ x_{3}^{\prime}=-6 x_{1}+20 x_{2}-15 x_{3}+16 & x_{3}(0)=-1 \end{array}\right. \]

8.2.2.3 Systems of differential equations

3.83 $

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

8.3.2 Systems of ordinary differential equations

2.55 $

Problem: Find the general solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{c} x^{\prime}=6 x-2 y-z+5 \\ y^{\prime}=-x+5 y-z+e^{t} \\ z^{\prime}=x-2 y+4 z+2 \end{array}\right. \]

8.3.3 Systems of ordinary differential equations

3.06 $

Problem: Find the particular solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{ll} \dot{x}=2 x+y, & x(0)=1 \\ \dot{y}=x+2 y, & y(0)=3 \end{array}\right. \]

8.3.4 Systems of ordinary differential equations

1.02 $

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

8.3.5 Systems of ordinary differential equations

1.53 $

Problem: Let's calculate the integral: 1) \( \int \frac{d x}{\sqrt{5-x^{2}}} \), 2) \( \int \frac{x+23}{x^{2}+x-20} d x \), 3) \( \int x e^{-7 x} d x \).

9.3.1.4 Indefinite integrals

2.55 $

Problem: Find indefinite integrals. Check the results by differentiation. 1) \( \int \frac{\tan x}{\cos ^{2} x} \), 2) \( \int \ln 3 x d x \), 3) \( \int \frac{x-8}{x(x-2)^{2}} d x \).

9.3.1.5 Indefinite integrals

2.55 $

(2) Problem: Calculate the improper integral or prove its divergence: \[ \int_{e}^{\infty} \frac{d x}{x \ln ^{2} x} \]

9.3.2.4 Improper integrals

1.28 $

3) Problem: Calculate the improper integral or prove its divergence: \[ \int_{-\infty}^{-3} \frac{x d x}{\left(x^{2}+1\right)^{2}} \]

9.3.2.5 Improper integrals

1.28 $

3) Problem: Calculate the definite integral: \[ \int_{5 / 2}^{10 / 3} \frac{\sqrt{x+2}+\sqrt{x-2}}{(\sqrt{x+2}-\sqrt{x-2})(x-2)^{2}} d x . \]

9.3.3.4 Definite Integrals

2.04 $

T Problem: Calculate the definite integral: \[ \int_{0}^{\sqrt{3}} \frac{d x}{\sqrt{x\left(4-x^{2}\right)^{3}}} \]

9.3.3.5 Definite Integrals

1.28 $

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