What is the solution for (d^2y)/(dt^2)+6(dy)/(dt)+9y=e^{-3t} ?

The solution for (d^2y)/(dt^2)+6(dy)/(dt)+9y=e^{-3t} is y=c_{1}e^{-3t}+c_{2}te^{-3t}+(e^{-3t}t^2)/2

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(d^2y)/(dt^2)+6(dy)/(dt)+9y=e^{-3t}

\frac{d ^{2} y}{d t ^{2}} + 6 \frac{d y}{d t} + 9 y = e^{- 3 t}

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

Solution steps

\frac{d ^{2} y}{d t ^{2}} + 6 \frac{d y}{d t} + 9 y = e^{- 3 t}

Solve linear ODE:

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

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

y^{^{''}} + 2 y^{^{'}} + y = \frac{1}{x} e^{- x}

y^{^{''}} + x + 4 y = 0

y^{^{''}} + 6 y^{^{'}} + 9 y = x e^{4 x}

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

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

What is the solution for (d^2y)/(dt^2)+6(dy)/(dt)+9y=e^{-3t} ?
The solution for (d^2y)/(dt^2)+6(dy)/(dt)+9y=e^{-3t} is y=c_{1}e^{-3t}+c_{2}te^{-3t}+(e^{-3t}t^2)/2