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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 3 y^{^{'}} + 2 y = 3 cos (3 t), y (0) = 2, y^{^{'}} (0) = 3

x^{^{''}} + x = sin (2 t)

y^{^{''}} + 16 y = 11 sec^{2} (4 t)

y^{^{''}} - 4 y^{^{'}} - 12 y = 2 sin (t)

12 y^{^{''}} + 29 y^{^{'}} - 11 y = - 9

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