What is the solution for y^{''}+y^'-6y=e^{-3x} ?

The solution for y^{''}+y^'-6y=e^{-3x} is y=c_{1}e^{2x}+c_{2}e^{-3x}-(e^{-3x}x)/5

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

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

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} - \frac{e ^{- 3 x} x}{5}

Solution steps

y^{''} (x) + y^{'} (x) - 6 y = e^{- 3 x}

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} - \frac{e ^{- 3 x} x}{5}

y = c_{1} e^{2 x} + c_{2} e^{- 3 x} - \frac{e ^{- 3 x} x}{5}

y^{^{''}} - 2 y^{^{'}} - 3 y = 12 e^{x} - 9

\frac{d ^{2} y}{d x ^{2}} + 4 y = sin (x)

y^{^{''}} + 16 y = cos (4 t)

4 y^{^{''}} - y = e^{\frac{x}{2}} + 4

y^{^{''}} + 4 y = 3 csc (2 x)

What is the solution for y^{''}+y^'-6y=e^{-3x} ?
The solution for y^{''}+y^'-6y=e^{-3x} is y=c_{1}e^{2x}+c_{2}e^{-3x}-(e^{-3x}x)/5