Initial value problem how would I start?

Question

I have this problem for a practice exam. Could someone tell me how to start on this problem?

Answer 1

The characteristic equation for this differential equation is $r^2+ 2\zeta\omega_n r+ \omega^2_n= 0$. By the quadratic formula, $r= \frac{-\zeta\omega_n\pm\sqrt{\zeta^2\omega^2_n- 4\omega^2_n}}{2}= -\omega_n\frac{\zeta\pm\sqrt{\zeta^2- 4}}{2}$. Since $\zeta< 1$, $\zeta^2- 4< 0$ and we can write that as $-\omega_n\left(\frac{\zeta}{2}\pm i\frac{\sqrt{4- \zeta^2}}{2}\right)$.

So the general solution to the differential equation is of the form $x(t)= e^{-\omega\zeta t/2}\left(C_1\cos\left(\omega_n\sqrt{4- \zeta^2}t/2\right)+ C_2\sin\left(\omega_n\sqrt{4- \zeta^2}t/2\right)\right)$.

It is easy to see that, since $x(0)= x_0$, that $C_1= x_0$. Differentiate and set t= 0 to use $x'(0)= 0$ to find $C_1$.