What is the solution for (d^2x)/(dt^2)+6(dx)/(dt)+5x=e^{-t} ?

The solution for (d^2x)/(dt^2)+6(dx)/(dt)+5x=e^{-t} is x=c_{1}e^{-t}+c_{2}e^{-5t}+(e^{-t}t)/4

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(d^2x)/(dt^2)+6(dx)/(dt)+5x=e^{-t}

\frac{d ^{2} x}{d t ^{2}} + 6 \frac{d x}{d t} + 5 x = e^{- t}

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

Solution steps

\frac{d ^{2} x}{d t ^{2}} + 6 \frac{d x}{d t} + 5 x = e^{- t}

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} + y = \frac{e ^{t}}{t ^{3}}

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

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

y^{^{'''}} + 4 y^{^{''}} + 5 y^{^{'}} + 2 y = 10 cos (t)

y^{^{''}} - y^{^{'}} - 6 y = 2 e^{2 x}

What is the solution for (d^2x)/(dt^2)+6(dx)/(dt)+5x=e^{-t} ?
The solution for (d^2x)/(dt^2)+6(dx)/(dt)+5x=e^{-t} is x=c_{1}e^{-t}+c_{2}e^{-5t}+(e^{-t}t)/4