Discussion

If F(t) is periodic, the solution can be determined by methods from MA 241:

For the homogeneous part:

If C^2>4MK, the homogeneous solution is u(t)=c1Exp[-(a-b)t]+c2Exp[-(a+b)t] ,
where a=C/(2M), b=1/(2M) Sqrt[C^2-4MK] (overdamping)

If C^2<4MK, the homogeneous solution is
u(t)=Exp[-at](Acos wt + Bsin wt), where a=C/(2M),
w=1/(2M) Sqrt[4MK-C^2] (underdamping)

If C^2=4MK, the homogeneous solution is
u(t)=(c1t+c2)Exp[-at] where a=C/(2M) (critical damping)

For the particular solution, you used the method of undetermined coefficients: assume up(t) is of the form acos wt + bsin wt and substitute in.

What if F(t) is not periodic? Represent F(t) as a Fourier series! Then it is made up of periodic parts! Solve the inhomogeneous differential equation Mu''+Cu'+Ku=each term of Fourier series. You will get a particular solution for each term in the series, and the solution to the problem with forcing term F(t) is the inhomogeneous solution plus the sum of the particular solutions.

Enough talk-let's see some action.

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