What is the solution for y^{''}+8y^'+16y=t^{-2}e^{-4t} ?

The solution for y^{''}+8y^'+16y=t^{-2}e^{-4t} is y=c_{1}e^{-4t}+c_{2}te^{-4t}-e^{-4t}ln(t)-e^{-4t}

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y^{''}+8y^'+16y=t^{-2}e^{-4t}

y^{^{''}} + 8 y^{^{'}} + 16 y = t^{- 2} e^{- 4 t}

y = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - e^{- 4 t} ln (t) - e^{- 4 t}

Solution steps

y^{''} (t) + 8 y^{'} (t) + 16 y = t^{- 2} e^{- 4 t}

Solve linear ODE:

y = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - e^{- 4 t} ln (t) - e^{- 4 t}

y = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - e^{- 4 t} ln (t) - e^{- 4 t}

What is the solution for y^{''}+8y^'+16y=t^{-2}e^{-4t} ?
The solution for y^{''}+8y^'+16y=t^{-2}e^{-4t} is y=c_{1}e^{-4t}+c_{2}te^{-4t}-e^{-4t}ln(t)-e^{-4t}