My class is solving the Cauchy-Euler differential equation a t^2 y'' + b t y' + c y = 0. The solutions are powers of $t$, $y = t^r$ and then you solve for $r$ using the characteristic equation $a r^2 + (b-a) r + c = 0$. This has two roots $r = \overline{r} \pm \Delta r$ and the general solution is $y = A t^{\overline{r} + \Delta r} + B t^{\overline{r} - \Delta r }$.
What happens if we get double roots, i.e. $(b-a)^2 = 4ac$ or $\Delta r = 0$? We guess a solution of $y = t^r \ln t$. Can derive this guess taking the limit of two distinct roots solution as $\Delta r$ tends to 0? In some sense $ A t^{\overline{r} + \Delta r} + B t^{\overline{r} - \Delta r } = t^{\overline{r}} ( A t^{ \Delta r} + B t^{ - \Delta r }) \to t^{\overline{r}} (C + D \ln t )$ for some constants $A,B,C,D$.
In a way, it is plausible the limit works this way because of the integral formula: $ \int t^{r - 1}\mathrm dt = \begin{cases} \frac{t^r}{r} & r \neq 0\\ \ln t & r = 0 \end{cases}$ And I expect this is coming into play in the Euler-Cauchy equation. How to make it rigorous, though?