What is the solution for y^{''''}-y=8e^x ?

The solution for y^{''''}-y=8e^x is y=c_{1}e^x+c_{2}e^{-x}+c_{3}cos(x)+c_{4}sin(x)+2xe^x

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y^{''''}-y=8e^x

y^{^{''''}} - y = 8 e^{x}

y = c_{1} e^{x} + c_{2} e^{- x} + c_{3} cos (x) + c_{4} sin (x) + 2 x e^{x}

Solution steps

y^{(4)} (x) - y = 8 e^{x}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} + c_{3} cos (x) + c_{4} sin (x) + 2 x e^{x}

y = c_{1} e^{x} + c_{2} e^{- x} + c_{3} cos (x) + c_{4} sin (x) + 2 x e^{x}

(D^{2} + 9) y = cos (2 x)

y^{^{''}} + y = t, y (0) = 4, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 3 y^{^{'}} - 2 y = 2 t^{2} + 4 t + 4

y^{^{''}} + 4 y^{^{'}} + 4 y = t^{2} - 2 t

y^{^{''}} - y^{^{'}} - 6 y = - 2 t sin (t)

What is the solution for y^{''''}-y=8e^x ?
The solution for y^{''''}-y=8e^x is y=c_{1}e^x+c_{2}e^{-x}+c_{3}cos(x)+c_{4}sin(x)+2xe^x