What is the solution for D^2(D+3)^2y=9e^{-3x} ?

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

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D^2(D+3)^2y=9e^{-3x}

D^{2} (D + 3)^{2} y = 9 e^{- 3 x}

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

Solution steps

D^{2} (D + 3)^{2} y = 9 e^{- 3 x}

Solve linear ODE:

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

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

y^{^{'''}} - 2 y^{^{''}} - y^{^{'}} + 2 y = e^{6 t}

2 y^{^{''}} + 3 y^{^{'}} + y = sin (x), y (0) = 1, y^{^{'}} (0) = 2

y^{^{''}} + 4 y^{^{'}} = 10 sin (2 x)

y^{^{''}} - 4 y^{^{'}} + 3 y = 1, y (0) = 1, y^{^{'}} (0) = 0

(D^{2} + 1) y = sin (x)

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