What is the solution for x^{''}+8x^'+16x=6sin(4t) ?

The solution for x^{''}+8x^'+16x=6sin(4t) is x=c_{1}e^{-4t}+c_{2}te^{-4t}-3/16 cos(4t)

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x^{''}+8x^'+16x=6sin(4t)

x^{^{''}} + 8 x^{^{'}} + 16 x = 6 sin (4 t)

x = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - \frac{3}{16} cos (4 t)

Solution steps

x^{''} (t) + 8 x^{'} (t) + 16 x = 6 sin (4 t)

Solve linear ODE:

x = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - \frac{3}{16} cos (4 t)

x = c_{1} e^{- 4 t} + c_{2} t e^{- 4 t} - \frac{3}{16} cos (4 t)

y^{^{'''}} + y^{^{'}} = - 2 sin (x)

x^{^{''}} + 16 x = 2 sin (t)

y^{^{''}} - 6 y^{^{'}} + 5 y = 9 e^{2 x}

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

y^{^{'''}} - 7 y^{^{'}} - 6 y = 26 e^{2 x}

What is the solution for x^{''}+8x^'+16x=6sin(4t) ?
The solution for x^{''}+8x^'+16x=6sin(4t) is x=c_{1}e^{-4t}+c_{2}te^{-4t}-3/16 cos(4t)