What is the solution for x^{''}+x=5cos(t) ?

The solution for x^{''}+x=5cos(t) is x=c_{1}cos(t)+c_{2}sin(t)+(5tsin(t))/2

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x^{''}+x=5cos(t)

x^{^{''}} + x = 5 cos (t)

x = c_{1} cos (t) + c_{2} sin (t) + \frac{5 t sin ( t )}{2}

Solution steps

x^{''} (t) + x = 5 cos (t)

Solve linear ODE:

x = c_{1} cos (t) + c_{2} sin (t) + \frac{5 t sin ( t )}{2}

x = c_{1} cos (t) + c_{2} sin (t) + \frac{5 t sin ( t )}{2}

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

y^{^{''}} + 2 y = \frac{1}{cos ( 2 x )}

y^{^{''}} - 4 y^{^{'}} + 4 y = t^{- 3} e^{2 t}

y^{^{''}} + 4 y^{^{'}} - 32 = 0

y^{^{''}} + 16 y = e^{5 t}, y (0) = 0, y^{^{'}} (0) = 0

What is the solution for x^{''}+x=5cos(t) ?
The solution for x^{''}+x=5cos(t) is x=c_{1}cos(t)+c_{2}sin(t)+(5tsin(t))/2