What is the solution for (d^2y)/(dx^2)-y=sin(x)+e^{-5x} ?

The solution for (d^2y)/(dx^2)-y=sin(x)+e^{-5x} is y=c_{1}e^x+c_{2}e^{-x}-1/2 sin(x)+1/24 e^{-5x}

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(d^2y)/(dx^2)-y=sin(x)+e^{-5x}

\frac{d ^{2} y}{d x ^{2}} - y = sin (x) + e^{- 5 x}

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{1}{2} sin (x) + \frac{1}{24} e^{- 5 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} - y = sin (x) + e^{- 5 x}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{1}{2} sin (x) + \frac{1}{24} e^{- 5 x}

y = c_{1} e^{x} + c_{2} e^{- x} - \frac{1}{2} sin (x) + \frac{1}{24} e^{- 5 x}

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

y^{^{''}} - 4 y^{^{'}} + 3 y = 1

y^{^{''}} + 9 y = 2 cos (3 x) + 2 sin (3 x)

y^{^{''}} - 32 y = 16 cos (x)

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 91

What is the solution for (d^2y)/(dx^2)-y=sin(x)+e^{-5x} ?
The solution for (d^2y)/(dx^2)-y=sin(x)+e^{-5x} is y=c_{1}e^x+c_{2}e^{-x}-1/2 sin(x)+1/24 e^{-5x}