What is the solution for x^{''}+5x^'+4x=8e^t ?

The solution for x^{''}+5x^'+4x=8e^t is x=c_{1}e^{-t}+c_{2}e^{-4t}+4/5 e^t

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x^{''}+5x^'+4x=8e^t

x^{^{''}} + 5 x^{^{'}} + 4 x = 8 e^{t}

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

Solution steps

x^{''} (t) + 5 x^{'} (t) + 4 x = 8 e^{t}

Solve linear ODE:

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

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

y^{^{''}} + 20 y^{^{'}} + 200 y = 1

y^{^{''}} + 20 y^{^{'}} + 200 y = 2

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

y^{^{''}} - y = 8 e^{9 x} + x e^{9 x} + 5 x^{2} e^{9 x}

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

What is the solution for x^{''}+5x^'+4x=8e^t ?
The solution for x^{''}+5x^'+4x=8e^t is x=c_{1}e^{-t}+c_{2}e^{-4t}+4/5 e^t