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Problem: Solve the system of differential equations, using the operational method: \[ \left\{\begin{array}{l} y^{\prime}+5 y-2 x=e^{t} \\ x^{\prime}-y+6 x=e^{-2 t}, \quad x(0)=0, \quad y(0)=0 . \end{array}\right. \]

8.2.2.4 Systems of differential equations

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Problem: Find the maximum value of the parameter \( a \), for which the system has a solution: \[ \left\{\begin{array}{l} 4 \sin ^{2} y-a=16 \sin ^{2} \frac{2 x}{7}+9\left(\tan ^{-1}\right) \frac{2 x}{7} \\ \left(\pi^{2} \cos ^{2} 3 x-2 \pi^{2}-72\right) y^{2}=2 \pi^{2}\left(1+y^{2}\right) \sin 3 x \end{array}\right. \]

12.3.1.5 Algebraic

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Problem: Two groups and a mapping between them are given. Is the given mapping a homomorphism, epimorphism, monomorphism, isomorphism? \[ \begin{array}{l} \left(Z_{3},+\right),\left(Z_{9},+\right), \\ f(\overline{0})=\overline{0}, \quad f(\overline{1})=\overline{3}, \quad f(\overline{2})=\overline{6} . \end{array} \]

1.6.22 Fields, Groups, Rings

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Problem: At what values of \( \lambda \) the following quadratic form is positive definite: \[ f=5 x_{1}^{2}+x_{2}^{2}+\lambda x_{3}^{2}+4 x_{1} x_{2}-2 x_{1} x_{3}-2 x_{2} x_{3} . \]

1.8.2 Quadratic forms

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Problem: Using the Lagrange method, bring the quadratic form \( x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1} \) to normal form and find the corresponding non-degenerate transformation of variables.

1.8.3 Quadratic forms

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Problem: \( \varphi- \) linear operator of rank 1. Prove that at least one of the operators \( \varphi+\varepsilon, \varphi-\varepsilon \) is invertible. ( \( \varepsilon- \) an identity operator).

1.7.7 Linear transformations

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Problem: Convert the quadratic form to canonical form \( 6 x_{1}^{2}+5 x_{2}^{2}+7 x_{3}^{2}-4 x_{1} x_{2}+4 x_{1} x_{3} \) and find the corresponding orthogonal transformation of the variables.

1.8.4 Quadratic forms

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Problem: For the following forms, find a non-degenerate linear transformation that takes the form \( f(x) \) in the form \( g(y) \) (the desired transformation is defined ambiguously). \[ \begin{array}{l} f(x)=2 x_{1}^{2}+9 x_{2}^{2}+3 x_{3}^{2}+8 x_{1} x_{2}-4 x_{1} x_{3}-10 x_{2} x_{3}, \\ g(y)=2 y_{1}^{2}+3 y_{2}^{2}+6 y_{3}^{2}-4 y_{1} y_{2}-4 y_{1} y_{3}+8 y_{2} y_{3} . \end{array} \]

1.8.5 Quadratic forms

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Problem: Find the normal form of the quadratic form using the Lagrange method: \[ f=x_{1}^{2}+x_{2}^{2}+3 x_{3}^{2}+4 x_{1} x_{2}+2 x_{2} x_{3} . \]

1.8.6 Quadratic forms

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Problem: Find out, does the set of all real numbers forms the linear space if the sum of any two elements \( a \) and \( b \) is defined in it, equal to \( a+b \) and the product of any element \( a \) by any real number \( \varepsilon \), equal to \( \varepsilon \cdot a \).

1.9.3 Linear spaces

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Problem: Prove that the set \( M \) of functions \( x(t) \), given on the area D, forms a linear space. Find its dimension and basis. \[ M=\{\alpha+\beta \tan t+\gamma \cot t\}, \quad t \in\left(0, \frac{\pi}{2}\right) . \]

1.9.4 Linear spaces

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Problem: Let \( V \) - linear space of all symmetric polynomials of degree at most two over \( \mathbb{R} \) from two variables \( x \) and \( y \). Choose a basis in space \( V \) and find the operator matrix \( L \) in this basis, if \[ L(f)(x, y)=(2 x+3 y) \frac{\partial f}{\partial x}+(3 x+2 y) \frac{\partial f}{\partial y} . \]

1.9.5 Linear spaces

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Problem: Prove that the set of vectors \( L=\left\{\bar{a}=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right) \mid \alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}=0\right\} \) is the subspace of the space \( R^{n} \), a sequence of \( n \)-dimensional vectors \( \bar{a}_{1}=(1,0, \ldots, 0,-1) \), \( \bar{a}_{2}=(0,1, \ldots, 0,-1), \ldots, \bar{a}_{n-1}=(0,0, \ldots, 1,-1) \quad \) basis of this subspace.

1.9.6 Linear spaces

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Problem: Prove that the set of \( n \)-dimensional vectors \( L=\{\bar{a}=\underbrace{(\alpha, \beta, \alpha, \beta, \ldots)}_{n} \mid \alpha, \beta \in R\} \quad \) is the subspace of the space \( \mathbb{R}^{n} \), find the basis and dimension of this subspace.

1.9.7 Linear spaces

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Problem: Let \( M \) be a set of polynomials \( P \in \mathrm{P}_{n} \) with real coefficients satisfying the specified conditions. Prove that \( M \) - linear subspace in \( \mathrm{P}_{n} \), find its basis and dimension. Complement basis \( M \) to the basis of the whole space \( P_{n} \). \[ n=3, \quad M=\left\{P \in \mathrm{P}_{3} \mid P^{\prime \prime}(1)+P^{\prime}(0)=0\right\} . \]

1.9.8 Linear spaces

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