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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{'''}} + y^{^{''}} = 5 x^{2}

y^{^{''}} - 4 y^{^{'}} + 4 y = t^{3}

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 9 y = (x + 1) e^{x}

y^{^{''}} - 4 y^{^{'}} + 2 y = 5 e^{x}

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

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