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?