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Problem: Solve the system of linear differential equations: \[ x^{\prime}=-\frac{2}{t} x, \quad y^{\prime}=y+\frac{t+2}{t} x . \]

8.3.23 Systems of ordinary differential equations

3 $

Problem: Find the solution of the system of equations: \[ x^{\prime \prime}=2 x+3 y, y^{\prime \prime}=4 x-2 y \text {. } \]

8.3.24 Systems of ordinary differential equations

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Problem: Find the resting points (equilibrium) of the system and investigate the stability of the system at these points: \[ \left\{\begin{array}{l} \frac{d y}{d t}=\operatorname{sh}(A x)+\frac{\pi}{2} \operatorname{sh}(B y) \\ \frac{d x}{d t}=\cos \left(\frac{\pi}{2} x\right)+c e^{y}-D \sec y \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \( A \) & \( B \) & \( C \) & \( D \) \\ \hline- & - & - & - \end{tabular}

8.4.1.5 Stability of the systems of equations

4.99 $

Problem: Find the resting point (equilibrium) of the system and investigate the stability of the system at this points: \[ \left\{\begin{array}{l} \frac{d y}{d t}=C \cdot \arcsin \frac{B x}{\sqrt{1+(B x)^{2}}}+A \tan (D y) \\ \frac{d x}{d t}=A \operatorname{th}(B y)+D x e^{-\frac{C}{D} x} \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline+ & + & - & - \\ \hline \end{tabular}

8.4.1.6 Stability of the systems of equations

4.99 $

Problem: Find the resting points (equilibrium) of the system and investigate the stability of the system at these points: \[ \left\{\begin{array}{l} \frac{d y}{d t}=C \cdot \arcsin \frac{B x}{\sqrt{1+(B x)^{2}}}+A \tan (D y) \\ \frac{d x}{d t}=A \cdot \operatorname{th}(B y)+D \cdot x e^{-\frac{C}{D} x} \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline- & + & - & + \\ \hline \end{tabular}

8.4.1.7 Stability of the systems of equations

6.24 $

Problem: Determine at what values of the parameter \( \alpha \) the resting point of the system is stable: \[ x^{\prime}=-3 x+\alpha y, \quad y^{\prime}=-\alpha x+y . \]

8.4.1.8 Stability of the systems of equations

2.5 $

Problem: Investigate the stability of the solution of the following system of equations: \[ x^{\prime}=x+y, \quad y^{\prime}=x-y, \quad x(0)=y(0)=0 . \]

8.4.1.9 Stability of the systems of equations

3.75 $

Problem: Investigate the stability of the solution of the following system of equations: \[ x^{\prime}=-2 x-3 y, y^{\prime}=x+y, x(0)=y(0)=0 . \]

8.4.1.10 Stability of the systems of equations

2.5 $

Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=x+2 y, y^{\prime}=-3 x+y \text {. } \]

8.4.1.11 Stability of the systems of equations

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Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=-2 x+\frac{1}{3} y, \quad y^{\prime}=-2 x+\frac{1}{2} y . \]

8.4.1.12 Stability of the systems of equations

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Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=-y, \quad y^{\prime}=x-2 y \text {. } \]

8.4.1.13 Stability of the systems of equations

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Problem: Investigate the stability of the solution of the equation \( \dot{x}=t(x-1), x(0)=1 \).

8.4.2.1 Stability of the equations

2.5 $

Problem: Let's investigate the stability of the solution of the equation \( \dot{x}=t(x-1), x(0)=-1 \).

8.4.2.2 Stability of the equations

2.5 $

Problem: Prove that in the space \( C[0,2] \) the following set is precompact \[ A=\left\{f \in C[0,2]|| f^{\prime \prime}(x)|\leq 3, \forall x \in[0,2] ;| f^{\prime}(0) \mid \leq 1, f(2)=3\right\} \text {. } \]

19.5.2 Compactness

3.75 $

Problem: Prove that in the space \( C[-1 ; 2] \) relatively compact (precompact) set \[ E=\left\{\varphi \in C[-1 ; 2],\left|\int_{-1}^{2} x \varphi(x) d x\right| \leq 3,\left|\varphi^{\prime}(x)\right| \leq 3\right\} . \]

19.5.3 Compactness

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