Higher order differential equations

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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

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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

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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

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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

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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

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Problem: Find the general solution of the inhomogeneous differential equation in higher derivatives: \[ y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=3 e^{-x}+\cos 2 x \text {. } \]

8.1.3.27 Higher order differential equations

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