Yesterday I solved this problem from Teschl's differential equations book:
Problem: (Paraphrased) Show that the change of variables
$$v = ue^r, \quad r = \int^z p(\zeta) d\zeta$$
transforms the equation
$$u'' + 2p(z)u' + q(z)u = 0$$
into the equation
$$v'' + \left[ q(z) - p(z) - p^2(z) \right]v = 0$$
Solution: Differentiating $v$ twice, and using the fact that $r' = p$, we have
\begin{align*} v'' & = (ue^r)'' \\ & = [(u' + pu) e^r]' \\ & = [(u' + pu)' + p(u' + pu)] e^r \\ & = [p^2u + p'u + 2pu' + u''] e^r \\ & = [p^2u + p'u - qu] e^r \\ & = [p^2 + p - q] v \end{align*}
which is what we wanted to show.
I cannot help but notice an analogy with the procedure for solving a quadratic equation (in characteristic different from 2):
Problem: Show that the change of variables
$$y = x + b$$
transforms the equation
$$x^2 + 2bx + c = 0$$
into the equation
$$y^2 - b^2 + c = 0$$
Solution: We have
$$y^2 = (x + b)^2 = x^2 + 2bx + b^2 = b^2 - c$$
which is what we wanted to show.
Is there some generalization of Galois theory that makes this analogy precise?