What is the solution for y^{''}-9y=9xe^{-3x} ?

The solution for y^{''}-9y=9xe^{-3x} is y=c_{1}e^{3x}+c_{2}e^{-3x}-(3e^{-3x}x^2)/4-(e^{-3x}x)/4

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y^{''}-9y=9xe^{-3x}

y^{^{''}} - 9 y = 9 x e^{- 3 x}

y = c_{1} e^{3 x} + c_{2} e^{- 3 x} - \frac{3 e ^{- 3 x} x ^{2}}{4} - \frac{e ^{- 3 x} x}{4}

Solution steps

y^{''} (x) - 9 y = 9 x e^{- 3 x}

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{- 3 x} - \frac{3 e ^{- 3 x} x ^{2}}{4} - \frac{e ^{- 3 x} x}{4}

y = c_{1} e^{3 x} + c_{2} e^{- 3 x} - \frac{3 e ^{- 3 x} x ^{2}}{4} - \frac{e ^{- 3 x} x}{4}

y^{^{'''}} - 2 y^{^{''}} + y^{^{'}} = x e^{x} + 5

y^{^{''}} - 64 y = sinh (8 x)

y^{^{''}} - y^{^{'}} - 2 y = - 2 x^{3} + x^{2} + 4 x + 1

y^{^{''}} - 5 y^{^{'}} + 4 y = x

y^{^{''}} - 4 y^{^{'}} + 3 y = x e^{2 x}

What is the solution for y^{''}-9y=9xe^{-3x} ?
The solution for y^{''}-9y=9xe^{-3x} is y=c_{1}e^{3x}+c_{2}e^{-3x}-(3e^{-3x}x^2)/4-(e^{-3x}x)/4