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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: \( \operatorname{ch} x, \operatorname{sh} x \).

8.1.3.6 Higher order differential equations

0.78 $

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: Investigate linear dependence for the following system of functions: \( 1, \sin x, \cos 2 x \).

8.1.3.10 Higher order differential equations

0.78 $

Problem: Knowing the fundamental system of solutions of the linear homogeneous differential equation, \( e^{2 x} \cos x, e^{2 x} \sin x \), write this equation. clip2net.com

8.1.3.11 Higher order differential equations

1.31 $

Problem: Knowing the fundamental system of solutions of the linear homogeneous differential equation, \( x^{3}, x^{4} \), write this equation. clip22net.com

8.1.3.12 Higher order differential equations

1.31 $

Problem: Knowing the fundamental system of solutions of the linear homogeneous differential equation, \( 2 x, x--2, e^{x}+1 \), write this equation.

8.1.3.18 Higher order differential equations

1.31 $

Problem: Knowing the fundamental system of solutions of the linear homogeneous differential equation, \( e^{2 x}, \sin x, \cos x \), write this equation. clip2net.com

8.1.3.19 Higher order differential equations

1.31 $

Problem: Knowing the fundamental system of solutions \( e^{x}, \sin x, \cos x \) of the linear homogeneous differential equation, find its particular solution that satisfies the initial conditions: \[ y(0)=3, y^{\prime}(0)=4, y^{\prime \prime}(0)=-1 \text {. } \]

8.1.3.20 Higher order differential equations

1.31 $

Problem: Checking that the functions \( y_{1}(x)=e^{x} \) and \( y_{2}(x)=x \) form a fundamental system of solutions of the equation \( y^{\prime \prime}-\frac{x}{x-1} y^{\prime}+\frac{1}{x-1} y=0 \), find the general solution of the equation \( (x-1) y^{\prime \prime}-x y^{\prime}+y=(x-1)^{2} \). clip2net.com

8.1.3.21 Higher order differential equations

2.09 $

Problem: Checking that the functions \( y_{1}(x)=\cos x \) and \( y_{2}(x)= \) \( =x \cos x \) form a fundamental system of solutions of the equation \( y^{\prime \prime}+2 \tan x \cdot y^{\prime}+\left(2 \tan ^{2} x+1\right) y=0 \), find the general solution of the equation \( \cot x \cdot y^{\prime \prime}+2 y^{\prime}+(2 \tan x+\cot x) y=\cos ^{2} x \). clip2net.com

8.1.3.22 Higher order differential equations

2.61 $

Problem: Find the general solution of the differential equation: \[ y^{(6)}-2 y^{(5)}+3 y^{(4)}-4 y^{\prime \prime \prime}+3 y^{\prime \prime}-2 y^{\prime}+y=0 \text {. } \] clip2net.com

8.1.3.23 Higher order differential equations

1.83 $

Problem: Find the particular solution of the equation \( y^{(4)}-y=8 e^{x} \), satisfying the initial conditions \( y(0)=0, y^{\prime}(0)=2, y^{\prime \prime}(0)=4, y^{\prime \prime \prime}(0)=6 \). clip2net.com

8.1.3.24 Higher order differential equations

1.83 $

Problem: Find the general solution of Euler's equation: \( x^{2} y^{\prime \prime \prime}-3 x y^{\prime \prime}+3 y^{\prime}=0 \). clip2net.com

8.1.3.26 Higher order differential equations

1.83 $

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