What is the solution for x^{''}+16x=8sin(2t) ?

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

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x^{''}+16x=8sin(2t)

x^{^{''}} + 16 x = 8 sin (2 t)

x = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{2}{3} sin (2 t)

Solution steps

x^{''} (t) + 16 x = 8 sin (2 t)

Solve linear ODE:

x = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{2}{3} sin (2 t)

x = c_{1} cos (4 t) + c_{2} sin (4 t) + \frac{2}{3} sin (2 t)

y^{^{'''}} - 3 y^{^{'}} - 2 y = 2 x + 1

y^{^{''}} + 4 y^{^{'}} + 3 y = x

y^{^{'''}} - 2 y^{^{''}} - 4 y^{^{'}} + 8 y = e^{2 x}

y^{^{''}} + 2 y^{^{'}} + y = 1, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 3 y^{^{'}} + 2 y = x e^{x}

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