What is the solution for y^{''}-4y=4xe^{-2x} ?

The solution for y^{''}-4y=4xe^{-2x} is y=c_{1}e^{2x}+c_{2}e^{-2x}-(e^{-2x}x^2)/2-(e^{-2x}x)/4

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y^{''}-4y=4xe^{-2x}

y^{^{''}} - 4 y = 4 x e^{- 2 x}

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{e ^{- 2 x} x ^{2}}{2} - \frac{e ^{- 2 x} x}{4}

Solution steps

y^{''} (x) - 4 y = 4 x e^{- 2 x}

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{e ^{- 2 x} x ^{2}}{2} - \frac{e ^{- 2 x} x}{4}

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{e ^{- 2 x} x ^{2}}{2} - \frac{e ^{- 2 x} x}{4}

What is the solution for y^{''}-4y=4xe^{-2x} ?
The solution for y^{''}-4y=4xe^{-2x} is y=c_{1}e^{2x}+c_{2}e^{-2x}-(e^{-2x}x^2)/2-(e^{-2x}x)/4