I have an ODE $$m\frac{d^2x}{dt^2} +c\frac{dx}{dt} + kx = r(t)$$
Where c is a dampening constant, k is considered a spring stiffness and r(t) is a force.
The first part of my question is to work out a homogeneous solution when m = 1, k = 144 and c = 40.
For this I assumed $$y = Ce^{rt}$$ and $$\frac{dx}{dt} = r.Ce^{rt}$$ and $$\frac{d^2x}{dt^2} = r^2.Ce^{rt}$$
My final homogeneous solution was
$$y_h = C_1e^{-4t} + C_2e^{-36t}$$
First off, if someone could check this and confirm whether it is correct or incorrect that would be very helpful.
Secondly, the next part is to find the particular solution given $$r(t) = 60 cos (4 π t)$$
I believe the format to answer this is in the form of $$Y_p = A cos (bt) + B sin (bt)$$ However I am not sure about this now how I go about finding the final particular solution anyway. If someone could help me through this that would be great.
Thank you.