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}

y^{^{''}} + 9 y = 10 e^{- t}, y (0) = 0, y^{^{'}} (0) = 0

y^{^{'''}} - y^{^{'}} = 5 sin (t)

y^{^{''}} + 4 y^{^{'}} + 8 y = sin (x), y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} + 3 y^{^{'}} + 33 y = 9 cos (5 x)

2 y^{^{''}} + 3 y^{^{'}} - 2 y = 14 x^{2} - 4 x - 11

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}