What is the solution for (D^2-1)y=xe^{3x} ?

The solution for (D^2-1)y=xe^{3x} is y=c_{1}e^x+c_{2}e^{-x}+(e^{3x}x)/8-3/32 e^{3x}

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(D^2-1)y=xe^{3x}

(D^{2} - 1) y = x e^{3 x}

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{3 x} x}{8} - \frac{3}{32} e^{3 x}

Solution steps

(D^{2} - 1) y = x e^{3 x}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{3 x} x}{8} - \frac{3}{32} e^{3 x}

y = c_{1} e^{x} + c_{2} e^{- x} + \frac{e ^{3 x} x}{8} - \frac{3}{32} e^{3 x}

y^{^{''}} - 2 y^{^{'}} + y = 15 e^{x}

y^{^{''}} - 2 y^{^{'}} + 10 y = e^{- x}

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

D (x) = 72 - 1.8 x

y^{^{'''}} - 2 y^{^{''}} + y^{^{'}} = (3 x - 4) e^{- 2 x} - 5 e^{x}

What is the solution for (D^2-1)y=xe^{3x} ?
The solution for (D^2-1)y=xe^{3x} is y=c_{1}e^x+c_{2}e^{-x}+(e^{3x}x)/8-3/32 e^{3x}