What is the solution for y^{''}+16y^'+64y=t^{-2}e^{-8t} ?

The solution for y^{''}+16y^'+64y=t^{-2}e^{-8t} is y=c_{1}e^{-8t}+c_{2}te^{-8t}-e^{-8t}ln(t)-e^{-8t}

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y^{''}+16y^'+64y=t^{-2}e^{-8t}

y^{^{''}} + 16 y^{^{'}} + 64 y = t^{- 2} e^{- 8 t}

y = c_{1} e^{- 8 t} + c_{2} t e^{- 8 t} - e^{- 8 t} ln (t) - e^{- 8 t}

Solution steps

y^{''} (t) + 16 y^{'} (t) + 64 y = t^{- 2} e^{- 8 t}

Solve linear ODE:

y = c_{1} e^{- 8 t} + c_{2} t e^{- 8 t} - e^{- 8 t} ln (t) - e^{- 8 t}

y = c_{1} e^{- 8 t} + c_{2} t e^{- 8 t} - e^{- 8 t} ln (t) - e^{- 8 t}

y^{^{'''}} + y^{^{''}} + y^{^{'}} + y = e^{- t} + 4 t

y^{^{''}} + 6 y^{^{'}} + 9 y = e^{3 x} + 2

y^{^{''}} + 4 y = t sin^{2} (t)

y^{^{''}} - 2 y^{^{'}} - 15 = 0

(D^{2} - 3 D + 2) y = x e^{x}

What is the solution for y^{''}+16y^'+64y=t^{-2}e^{-8t} ?
The solution for y^{''}+16y^'+64y=t^{-2}e^{-8t} is y=c_{1}e^{-8t}+c_{2}te^{-8t}-e^{-8t}ln(t)-e^{-8t}