# Substitution in ODE

A tutorial (page 17) on ODE scaling takes the equation $m\ddot{u}+c\dot{u}+ku=F_0sin(\omega t)$, with $u$ a function of $t$, and substitutes $\eta=u/a$, $\tau=t/b$ to get:

$$\frac{ma}{b^2}\ddot{\eta}+\frac{ca}{b}\dot{\eta}+ka\eta=F_0sin(\omega\tau)$$

Now, when I differentiate $\eta=u/a$, I get $\dot{\eta}=t\dot{u}/a=\tau b\dot{u}/a$. Substituting, the final result is:

$$\frac{ma}{\tau^2b^2}\ddot{\eta}+\frac{ca}{\tau b}\dot{\eta}+ka\eta=F_0sin(\omega\tau b)$$

That's not the same. What am I missing?

Gerry has already explained the differences on the left-hand side. On the right-hand side, this is a typo in the book. The next equation has $\sin\omega'\tau$ instead of $\sin\omega\tau$, with $\omega'=\omega\sqrt{m/k}$ and $b=\sqrt{m/k}$, so this is $\sin\omega\tau b$ as you expected. So it should be either $\sin\omega t$ or $\sin\omega'\tau$ on the right-hand side.
Differentiating $\eta=u/a$ gives $\eta'=u'/a$, not $\eta'=tu'/a$, no?
• @Don, I'm a little puzzled myself as to how they get $\omega\tau$ on the right side, where your $\omega\tau b$ looks better to me. Sorry, don't have the time to work through it now, but I'm sure someone will be along to help out. Aug 24, 2011 at 9:22