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

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

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x^{''}+x=acos(t)

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 36 y = 36 sec^{2} (6 x)

y^{^{'''}} + 25 y^{^{'}} = 10 sin (5 t) - 4 t^{2}

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

y^{^{''}} + 2 y^{^{'}} + y = 4000 sec (40 x)

a x^{^{''}} + b x^{^{'}} + c = 0

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