Chapter 1.8, Problem 13E

Consider the nonhomogeneous linear equation $\frac{dy}{dt}+2y=\cos 3t$ .
To find a particular solution, it is pretty clear that our guess must contain a cosinefunction, but it is not so clear that the guess must also contain a sine function.
(a) Guess $y_p\left(t\right)=\alpha \cos 3t$ and substitute this guess into the equation. Is there avalue of
$\alpha$ such that $y_p\left(t\right)$ is a solution?
(b) Write a brief paragraph explaining why the proper guess for a particular solution is $y_p\left(t\right)=\alpha \cos 3t+\beta \sin 3t$ .