Equation of Motion of a particle

Question

I've tried this question over and over, and I'm getting nowhere. I've even tried looking for a solution to help make sense of how to get there, but I've had no luck. Can anyone help me please?

A particle of mass $m$ moves in a straight line. At time $t$, its displacement from a fixed point, $O$, of the line is $x(t)$.

$(i)$ If the motion of the particle can be described by $$\frac{d^2x}{dt^2}\ + 2k\frac{dx}{dt}\ + 10k^2x=0$$ with $k>0$ and initial conditions $x(0)=0, \dfrac{dx}{dt}\Big|_{t=0}=u$, find an expression for $x(t)$.

$(ii)$ Show that when the particle is next at $O$ (i.e., when $x = 0$), we have $$\frac{dx}{dt}\ = -ue^{-\frac{\pi}{3}} $$

Answer 1

Assume $x$ is of the form $x(t)=e^{rt}$. Substitution into your ODE yields

$$r^2e^{rt}+2kre^{rt}+10k^2e^{rt} = e^{rt}(r^2+2kr+10k^2)=0$$

Since $e^{rt}\neq 0$, we have that $r^2+2kr+10k^2=0$, implying that there are two solutions for $r$, namely $$r_1=-k+3ik$$ $$r_2=-k-3ik$$

The general solution for an ODE with two complex roots, in this case, will take the form

$$x(t)=e^{-kt}(C_1\cos(3kt)+C_2\sin(3kt))$$

You said that at $t=0$ we have $x=0$. Therefore,

$$x(0)=e^{0}(C_1\cos(0)+C_2\sin(0))=C_1=0$$ Then,

$$x(t)=C_2e^{-kt}\sin(3kt)$$

Which implies that

$$\dfrac{dx}{dt}=3kC_2e^{-kt}\cos(3kt)-kC_2e^{-kt}\sin(3kt)$$

And with your condition of $\dfrac{dx}{dt}=u$ at $t=0$ lets us obtain

$$\dfrac{dx}{dt}\Bigg|_{t=0}=3kC_2e^{0}\cos(0)-kC_2e^{0}\sin(0)=3kC_2=u$$

Thus, $C_2=\dfrac{u}{3k}$ and our solution is

$$x(t)=\dfrac{u}{3k}e^{-kt}\sin(3kt)$$

Other than $t=0$, the next time that $x(t)=0$ will be $t=\dfrac{\pi}{3k}$. Then, we evaluate $\dfrac{dx}{dt}$ at $t=\dfrac{\pi}{3k}$.

$$\begin{align} \dfrac{dx}{dt}&=ue^{-kt}\cos(3kt)-\dfrac{u}{3}e^{-kt}\sin(3kt) \\ \dfrac{dx}{dt}\Bigg|_{t=\frac{\pi}{3k}}&=ue^{-k(\frac{\pi}{3k})}\cos(3k\dfrac{\pi}{3k}))-\dfrac{u}{3}e^{-k(\frac{\pi}{3k})}\sin(3k(\frac{\pi}{3k})) \\ &=ue^{-\frac{\pi}{3}}\cos(\pi)-\dfrac{u}{3}e^{-\frac{\pi}{3}}\sin(\pi) \\ &=-ue^{-\frac{\pi}{3}} \end{align}$$