What is the solution for y^{''}+4y^'+5y=5x+e^{-x} ?

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

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y^{''}+4y^'+5y=5x+e^{-x}

y^{^{''}} + 4 y^{^{'}} + 5 y = 5 x + e^{- x}

y = e^{- 2 x} (c_{1} cos (x) + c_{2} sin (x)) + x - \frac{4}{5} + \frac{1}{2} e^{- x}

Solution steps

y^{''} (x) + 4 y^{'} (x) + 5 y = 5 x + e^{- x}

Solve linear ODE:

y = e^{- 2 x} (c_{1} cos (x) + c_{2} sin (x)) + x - \frac{4}{5} + \frac{1}{2} e^{- x}

y = e^{- 2 x} (c_{1} cos (x) + c_{2} sin (x)) + x - \frac{4}{5} + \frac{1}{2} e^{- x}

y^{^{'''}} + 12 y^{^{''}} - 256 y = e^{- 8 x}

8 y^{^{''}} + 3 y^{^{'}} = 12 x

y^{^{''}} + y^{^{'}} - 6 y = e^{2 t}

y^{^{''}} - 4 y^{^{'}} + 5 y = (x + 1)^{3}

y^{^{''}} + 9 y = 18 t, y (\frac{π}{2}) = 0

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