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Problem: Find the general integral of the differential equation: \[ x \sqrt{4+y^{2}} d x+y \sqrt{1+x^{2}} d y=0 . \]

8.1.1.12 First order differential equations

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Problem: Find a solution to Cauchy problem: \[ y^{\prime}+\frac{y}{x}=3 x, \quad y(1)=1 . \]

8.1.1.14 First order differential equations

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Problem: Find the general solution of the differential equation: \[ \left(1+x^{2}\right) d y-2 x y d x=0 . \]

8.1.1.18 First order differential equations

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Problem: Investigate linear dependence for the following system of functions: \( x, 2 x, x^{2} \).

8.1.3.4 Higher order differential equations

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Problem: Investigate linear dependence for the following system of functions: \( e^{x}, x e^{x}, x^{2} e^{x} \).

8.1.3.5 Higher order differential equations

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Problem: Investigate linear dependence for the following system of functions: \( e^{x}, e^{x+1} \). clip2net.com

8.1.3.7 Higher order differential equations

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Problem: Investigate linear dependence for the following system of functions: \( x, 0, e^{x} \).

8.1.3.8 Higher order differential equations

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Problem: Write the differential equation of the hyperbola \( x^{2}-y^{2}=2 a x \).

8.1.4.4 Geometric and physical applications

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Problem: Write a differential equation of the family of curves, in which the segment of any tangent, closed between the coordinate axes, is divided by the tangent point \( M(x, y) \) in relation to \( |A M|:|M B|=2: 1 \), where \( A \) is the point of intersection of the tangent with the \( O y \) axis, \( B \) is the point of intersection with the \( O x \) axis. clip2net.com

8.1.4.5 Geometric and physical applications

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Problem: Using the method of isocline make the approximate family of integral curves of the equation \( y^{\prime}=x+y \). clip2net.com

8.1.4.6 Geometric and physical applications

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Problem: It's known that for independent random variables \( X_{1}, \cdots, X_{4} \) their expected values are \( E\left(X_{i}\right)=-2 \), the dispersions are \( D\left(X_{i}\right)=1 \), \( i=1, \cdots, 4 \). Find the dispersion of the product \( D\left(X_{1} ; \cdots X_{4}\right) \).

15.2.36 One dimensional random variables and their characteristics

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Problem: The random variable \( X \) is uniformly distributed on the segment \( [-4,2] \). Find \( E\left(e^{4 X}\right) \).

15.2.39 One dimensional random variables and their characteristics

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Problem: In case of what \( a \) the equation \( -10 a-10|a-x|-85=-15 x-x^{2} \) has at least one root in an interval \( (3 ; 7) \) ?

17.34 USE problems

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Problem: In case of what values of the parameter \( a \) the equation \( -4 a+4|a-x|+51=11 x-x^{2} \) has at least one solution in an interval \( (6 ; 9) \) ?

17.35 USE problems

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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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