# Thread: Modelling with second order ODEs

1. ## Modelling with second order ODEs

Show that for the overdamped system:

$\displaystyle \dfrac{d^2x}{dt^2}+3\dfrac{dx}{dt}+2x=0$

if $\displaystyle x=0$ and $\displaystyle \dfrac{dx}{dt} \neq 0$ when $\displaystyle t=0$ then x can never equal 0 again.

This is what I've done so far.

The characteristic equation is:

$\displaystyle \lambda^2+3\lambda+2=0$

$\displaystyle (\lambda+2)(\lambda+1)=0$

$\displaystyle \Rightarrow \lambda=-1, -2$

$\displaystyle x=Ae^{-2t}+Be^{-t}$

$\displaystyle \dfrac{dx}{dt}=-2Ae^{-2t}-Be^{-t}$

When $\displaystyle t=0, x=0$

$\displaystyle A+B=0$

When $\displaystyle t=0, \dfrac{dx}{dt}\neq 0$

$\displaystyle -2A-B\neq 0$

Not too sure if I'm doing it right, or what to do next?

2. What you've done so far is fine. You've also shown that since $\displaystyle A+B=0$ then $\displaystyle B = -A$.

This gives $\displaystyle x = Ae^{-2t} - Ae^{-t}$

$\displaystyle x = Ae^{-t}(e^{-t} - 1)$.

IF $\displaystyle x = 0$ then $\displaystyle 0 = Ae^{-t}(e^{-t} - 1)$

$\displaystyle Ae^{-t} = 0$ or $\displaystyle e^{-t} - 1 = 0$

$\displaystyle e^{-t} = 0$ or $\displaystyle e^{-t} = 1$

The first case is impossible. So the ONLY time that this can happen is $\displaystyle t = 0$.