Question

Consider a pendulum/spring modeled by the following equation: x"(t)+cx'(t)+x(t)=cos(t) where c>= 0 is the unspecified damping coefficient. Determine the amplitude of the steady periodic portion of the solution as a function of c and find the maximal amplitude. To what type of system does the maximal amplitude correspond?
If anyone can at least get me started on this that would be much appreciated. About to head off to work so ill check back when I'm done. Thanks.

Answer 1

My preferred method here would be to substitute
\[y =Ae^{it} \]

Into your differential equation and solve for A, the amplitude. Your result will be a complex function, so you will want to find its modulus:
\[ |A| = \sqrt{A^* A} \]
where \[A^* \] is the complex conjugate (found by replacing i with -i).

If you don't want to use complex numbers, then you can substitute
\[ y = A\cos(t + \delta) \]
where delta is some constant into the equation, but you'll need to do some tricky manipulation with sines and cosines and all that...