What is the solution for y^{''}+2y^'-8y=12e^{2x} ?

The solution for y^{''}+2y^'-8y=12e^{2x} is y=c_{1}e^{2x}+c_{2}e^{-4x}+2xe^{2x}

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y^{''}+2y^'-8y=12e^{2x}

y^{^{''}} + 2 y^{^{'}} - 8 y = 12 e^{2 x}

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

Solution steps

y^{''} (x) + 2 y^{'} (x) - 8 y = 12 e^{2 x}

Solve linear ODE:

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

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

y^{^{''}} - 5 y^{^{'}} - 6 y = - 3 x^{2}

y^{^{''}} - 2 y^{^{'}} + y = e^{2 x}, y (0) = 2, y^{^{'}} (0) = 0

y^{^{''}} + 4 y^{^{'}} + 9 y = x^{2} + 3 x

- y^{^{''}} + y^{^{'}} + 2 y = 72 t + 400 sin (t) + 36

y^{^{''}} - y^{^{'}} + y = 9 sin (t) - 21 e^{2 t}

What is the solution for y^{''}+2y^'-8y=12e^{2x} ?
The solution for y^{''}+2y^'-8y=12e^{2x} is y=c_{1}e^{2x}+c_{2}e^{-4x}+2xe^{2x}