What is the solution for 4y^{''}-16y= 1/(e^x) ?

The solution for 4y^{''}-16y= 1/(e^x) is y=c_{1}e^{2x}+c_{2}e^{-2x}-1/12 e^{-x}

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4y^{''}-16y= 1/(e^x)

4 y^{^{''}} - 16 y = \frac{1}{e ^{x}}

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{1}{12} e^{- x}

Solution steps

4 y^{''} (x) - 16 y = \frac{1}{e ^{x}}

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{1}{12} e^{- x}

y = c_{1} e^{2 x} + c_{2} e^{- 2 x} - \frac{1}{12} e^{- x}

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

y^{^{''}} + 11 y^{^{'}} + 28 y = - 2 e^{2 x} + 1

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

4 y^{^{''}} + 9 y = 15

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

What is the solution for 4y^{''}-16y= 1/(e^x) ?
The solution for 4y^{''}-16y= 1/(e^x) is y=c_{1}e^{2x}+c_{2}e^{-2x}-1/12 e^{-x}