What is the solution for y^{''}+8y^'+16y=8sin(4t) ?

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

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

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

y = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - \frac{1}{4} cos (4 t)

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - \frac{1}{4} cos (4 t)

y = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - \frac{1}{4} cos (4 t)

(4 D^{2} - 4 D + 1) y = 100

y^{^{''}} + 2 y^{^{'}} + 4 y = x e^{x}

y^{^{''}} + 5 y^{^{'}} + 6 y = 10 (1 - x) e^{- 2 x}

y^{^{''}} + 25 y = 3, y (0) = 1, y (0) = 5

y^{^{''}} + 6 y^{^{'}} + 8 y = 2 t - 3 cos (3 t)

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