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Problem: Find the general solution of the differential equation: \[ \sqrt{y} d x+\sqrt{x} d y=0 . \]

8.1.1.73 First order differential equations

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Problem: Find the solution of the differential equation: \[ \sqrt{x} d y=\sqrt{y} d x \text {. } \]

8.1.1.76 First order differential equations

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Problem: Find the general solution of the differential equation: \[ y^{\prime}-x y^{2}=0 \text {. } \]

8.1.1.89 First order differential equations

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Problem: Determine the type of the second order equation: \[ u_{x x}+2 u_{x y}-2 u_{y z}+2 u_{y y}+2 u_{z z}=0 . \]

11.5.4.5 With constant coefficients

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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: 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: 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: Find the coordinates of the vector $ X $ in the basis $ e^{\prime}=\left\{e_{1}^{\prime} ; e_{2}^{\prime} ; e_{3}^{\prime}\right\} $, if $ X=(1 ;-4 ; 8) $ in the basis $ e=\left\{e_{1} ; e_{2} ; e_{3}\right\} $ and \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}-3 e_{3} \\ e_{2}^{\prime}=\frac{3}{4} e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]

1.1.6 Vector Algebra

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Problem: Determine the type of the system of linear partial differential equations of the first order: \[ \left\{\begin{array}{l} 3 u_{x}+4 u_{y}-v_{y}+v_{x}=0 \\ u_{x}+v_{y}-v_{x}=0 \end{array}\right. \]

11.6.3 Systems of equations in partial derivatives of the first order

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Problem: Let the random variable $ \xi $ be the sum of the amount of the loan, requested by the bank's clients per day. This amount can change in the range from 0 to 1000000 (USD). The bank immediately withdraws the commissions of $ 1 \% $. What function will describe the amount of commissions, received by the bank, depending on the requested loan amount (the random variable $ \tau $ )? Find the distribution function of the random variable $ F_{\tau}(z) $.

15.2.19 One dimensional random variables and their characteristics

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Problem: Find out whether the set of integers divisible by 5 , form a group with respect to the addition. \[ M=\{n \in \mathbb{Z} \mid n: 5\}=\{5 k \mid k \in \mathbb{Z}\} . \]

1.6.23 Fields, Groups, Rings

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Problem: Find out whether the set of powers of number 7 with integer exponents form a group with respect to multiplication. \[ M=\left\{7^{n} \mid n \in \mathbb{Z}\right\} \text {. } \]

1.6.29 Fields, Groups, Rings

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Problem: Prove the statement by mathematical induction: $ \left(n^{5}-n\right) $ is a multiple of 5 for all natural $ n $.

6.6.15 Combinatorics

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Problem: Find the sequence $ \left\{a_{n}\right\} $, that satisfies the recurrence relation $ 2 \cdot a_{n+2}+10 \cdot a_{n+1}+8 \cdot a_{n}=0 $ and the initial conditions $ a_{1}=3, a_{2}=9 $.

6.6.18 Combinatorics

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