I am finding only a particular solution of the equation
$$y''+2\gamma y'+\Omega^2y=F\cos(\omega t)$$
which I do by assuming a solution of the form $Ae^{i\omega t}$, solving the associated equation for $A$, and then finding the real part. At the end of that process I obtain that
$$A = \frac{F}{\Omega^2-\omega^2+2i\omega\gamma}$$
which I suspect is correct. To find the real part of $Ae^{i\omega t}$ then involves a fairly hairy complex division problem, but I do eventually get the solution as this monster:
$$y=\frac{F\left([\Omega^2-\omega^2]\cos \omega t + [2\omega\gamma]\sin\omega t\right)}{(\Omega^2-\omega^2)^2+4\omega^2\gamma^2}$$
I next want to find the magnitude at which this reaches a maximum amplitude, and this is where I start to kind of lose spirits. I know I could use the formula $B\cos x+C\sin x = \sqrt{B^2+C^2}\sin (x+\alpha)$ to express this as a single sinusoid with its magnitude explicit. But that seems like a lot of work given how messy this already is. Is there a simpler solution?
In particular, I'm wondering: Can I just try to maximize $A$? If so, how? Do I minimize the denominator by taking the derivative of its magnitude and setting it to 0? All of the sudden that too sounds less appealing. Any suggestions would be appreciated.
(Also, if the math here got bad because I made a mistake, that'd be great to know too. So far I'm missing the reason why the prof chose to call the damping coefficient $2\gamma$ for instance, which makes me think I'm missing something or did something wrong.)