What is the solution for y^{''}-4y^'-12y=10e^{-2x} ?

The solution for y^{''}-4y^'-12y=10e^{-2x} is y=c_{1}e^{6x}+c_{2}e^{-2x}-(5e^{-2x}x)/4

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y^{''}-4y^'-12y=10e^{-2x}

y^{^{''}} - 4 y^{^{'}} - 12 y = 10 e^{- 2 x}

y = c_{1} e^{6 x} + c_{2} e^{- 2 x} - \frac{5 e ^{- 2 x} x}{4}

Solution steps

y^{''} (x) - 4 y^{'} (x) - 12 y = 10 e^{- 2 x}

Solve linear ODE:

y = c_{1} e^{6 x} + c_{2} e^{- 2 x} - \frac{5 e ^{- 2 x} x}{4}

y = c_{1} e^{6 x} + c_{2} e^{- 2 x} - \frac{5 e ^{- 2 x} x}{4}

y^{^{''}} + 8 y = 8 x

y^{^{'''}} + 12 y^{^{''}} + 48 y^{^{'}} + 64 y = 8 e^{- 4 x}

y^{^{''}} + y = 8 e^{- 6 x}, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - 6 y^{^{'}} - 6 y = 2 x e^{- x}

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

What is the solution for y^{''}-4y^'-12y=10e^{-2x} ?
The solution for y^{''}-4y^'-12y=10e^{-2x} is y=c_{1}e^{6x}+c_{2}e^{-2x}-(5e^{-2x}x)/4