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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

0 $

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

0 $

Problem: What is the parameter \( a \), if the equation \( \left|x^{2}-25\right|=6|x-a|-12 a \) has exactly two solutions?

17.36 USE problems

3.82 $

Problem: In case of what values of the parameter \( p \) the equation \( |x-16|=6|x-p|+2 p \) has exactly three solutions?

17.37 USE problems

2.55 $

Problem: Find the values of the parameter \( a \), for which the equation has exactly 3 solutions. \[ \left|x^{2}-25\right|=8|x-a|-6 a \text {. } \]

17.38 USE problems

3.82 $

Problem: Find the values of the parameter \( a \), for which the equation has exactly 3 roots: \[ \left|x^{2}-16\right|=6|x-a|+2 a \text {. } \]

17.39 USE problems

3.06 $

Problem: Find all values of the parameter \( a \), for which the equation \( \left|x^{2}-36\right|=8|x-a|-10 a \) has exactly one solution.

17.40 USE problems

2.55 $

Problem: Find all values of the parameter \( a \), for which the smallest value of the function \( y=4 a x+\left|x^{2}-8 x+7\right| \) is greater than 1 .

17.41 USE problems

2.04 $

Problem: Solve the Cauchy problem using Duhamel's formula: \( y^{\prime \prime}+4 y=2 \tan x, y(0)=0, y^{\prime}(0)=0 \).

8.2.1.6 Differential equations

3.82 $

Problem: Solve the Cauchy problem, using Duhamel's formula: \( y^{\prime \prime}+4 y=2 \tan x, y(0)=1, y^{\prime}(0)=2 \).

8.2.1.7 Differential equations

3.82 $

Problem: Solve the equation: \[ \int_{0}^{t} \operatorname{ch}(t-\tau) x(\tau) d \tau=\operatorname{ch} t-\cos t \] (with respect to the function \( x(t) \) ).

8.2.1.8 Differential equations

2.04 $

Problem: Solve the Cauchy problem for the system of linear differential equations, using the operational method: \[ \begin{array}{l} 2 x^{\prime \prime}+x-y^{\prime}=-3 \sin t, x+y^{\prime}=-\sin t, \\ x(0)=0, x^{\prime}(0)=1, y(0)=0 . \end{array} \]

8.2.2.5 Systems of differential equations

2.55 $

Problem: Solve the system of linear differential equations: \[ \frac{d y}{d x}=-\frac{y}{z}+x z, \quad \frac{d z}{d x}=-\frac{2 y}{x^{3}}+\frac{z}{x} . \]

8.3.20 Systems of ordinary differential equations

3.06 $

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

8.3.21 Systems of ordinary differential equations

2.55 $

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

8.3.22 Systems of ordinary differential equations

2.04 $

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