What is the solution for y^{''}+6y^'+9y=5e^{-3t} ?

The solution for y^{''}+6y^'+9y=5e^{-3t} is y=c_{1}e^{-3t}+c_{2}te^{-3t}+(5e^{-3t}t^2)/2

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y^{''}+6y^'+9y=5e^{-3t}

y^{^{''}} + 6 y^{^{'}} + 9 y = 5 e^{- 3 t}

y = c_{1} e^{- 3 t} + c_{2} t e^{- 3 t} + \frac{5 e ^{- 3 t} t ^{2}}{2}

Solution steps

y^{''} (t) + 6 y^{'} (t) + 9 y = 5 e^{- 3 t}

Solve linear ODE:

y = c_{1} e^{- 3 t} + c_{2} t e^{- 3 t} + \frac{5 e ^{- 3 t} t ^{2}}{2}

y = c_{1} e^{- 3 t} + c_{2} t e^{- 3 t} + \frac{5 e ^{- 3 t} t ^{2}}{2}

4 y^{^{''}} - 4 y^{^{'}} + y = 24 e^{\frac{t}{2}}

y^{^{''''}} + 36 y^{^{''}} = sin (6 t) + t e^{7 t} + 4

y^{^{''}} + 8 y = 16 sin (x)

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

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

What is the solution for y^{''}+6y^'+9y=5e^{-3t} ?
The solution for y^{''}+6y^'+9y=5e^{-3t} is y=c_{1}e^{-3t}+c_{2}te^{-3t}+(5e^{-3t}t^2)/2