This is the problem given:
I am not entirely sure what my Professor expected from an answer, but it seems I am to find the coefficient, angular frequency, and phase of the non-homogenous solution only.
I know that the steady state means $\frac{dy}{dt}=0$, but how can I know the initial condition for $y$?
Using method of undetermined coefficients I got $$y=c_1\exp\left(-\frac{2}{3}+\frac{\sqrt{19}}{3}\right)t+c_2\exp\left(-\frac{2}{3}-\frac{\sqrt{19}}{3}\right)-\frac{68}{353}\sin(2t)+\frac{32}{353}\cos(2t)$$
for the solution.
Is the answer simply $$A= \frac{32}{353},\, B=2,\, C=0?$$
Also is this what the solution should look like?
The blue line is the solution y and the green dashed line is the signal $4cos(2t)$
Best Answer
You are probably expected to use the fact that $$ a\cos x+b\sin x=\sqrt{a^2+b^2}\cos(x-\varphi), $$ where $\varphi$ satisfies $$ \cos \varphi=\frac{a}{\sqrt{a^2+b^2}},\\ \sin \varphi=\frac{b}{\sqrt{a^2+b^2}}. $$