Use Laplace Transform to solve the problem $y''+y'+y=u_2(t)e^{-t/2}\cos(\sqrt{3}t/2)$

Question

Use Laplace Transform to solve the problem

$y''+y'+y=u_2(t)e^{-t/2}\cos(\sqrt{3}t/2)$

$y(0)=0,y'(0)=1$

Answer 1

The thing that would make this problem difficult for my students is that the forcing function contains $u_2$ but the rest of the expression is not in terms of $t-2$. If they rewrote it as

$$u_2(t)e^{\frac{t-2}{2}+1}\cos\left(\frac{\sqrt{3}(t-2)}{2}+\sqrt{3}\right),$$

then used the sum of angles formula:

$$u_2(t)e^{\frac{t-2}{2}}e\left(\cos\left(\frac{\sqrt{3}(t-2)}{2}\right)\cos(\sqrt{3})-\sin\left(\frac{\sqrt{3}(t-2)}{2}\right)\sin\sqrt{3}\right),$$

then the formulae would be in their short table of Laplace transforms. The $e$, $\cos\sqrt{3}$ and $\sin\sqrt{3}$ are just distractions at this point.