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

The solution for (d^2x)/(dt^2)+2(dx)/(dt)+x=6e^{-t} is x=c_{1}e^{-t}+c_{2}te^{-t}+3t^2e^{-t}

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

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

x = c_{1} e^{- t} + c_{2} t e^{- t} + 3 t^{2} e^{- t}

Solution steps

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

Solve linear ODE:

x = c_{1} e^{- t} + c_{2} t e^{- t} + 3 t^{2} e^{- t}

x = c_{1} e^{- t} + c_{2} t e^{- t} + 3 t^{2} e^{- t}

What is the solution for (d^2x)/(dt^2)+2(dx)/(dt)+x=6e^{-t} ?
The solution for (d^2x)/(dt^2)+2(dx)/(dt)+x=6e^{-t} is x=c_{1}e^{-t}+c_{2}te^{-t}+3t^2e^{-t}