Time when $A =\frac{ A_0}{2}$

Question

I have solved a damped oscillator equation to obtain the solution $$x(t)=A_0e^{-t}\cos(\sqrt{3}t)$$ What method would be used to find the value of $t$ where $x(t)=\frac{A_0}{2}$?

Answer 1

You want the point of intersection for $$y= 2\cos (\sqrt{3} t)$$ and $$y=e^t$$

You may use a $TI84$ calculator to graph and find the intersection for you.

I found $t=0.41257682...$

Answer 2

Use the Newton–Raphson formula below to find the successive approximate solution,

$$t_{n+1} = t_n + \frac{2\cos\sqrt3 t_n -e^{t_n}}{2\sqrt3 \sin\sqrt3 t_n+e^{t_n}}$$

Use the starting guess, say, $t_0= 0.5$, to get $t_1=0.412597$, $t_2=0.412577$, ...