Cauchy–Euler equation

Remarks

The substitution $x = e^{t}$, $u(t) = y(e^t)$ changes it into a linear equation with constant coefficients for $u$. For $n = 2$, the trial solution $y = x^m$ gives the indicial equation $a_2 m (m - 1) + a_1 m + a_0 = 0$. For order $2$: a double root $m$ gives the solutions $x^m$ and $x^m \ln x$. Roots $\alpha \pm i \beta$ with $\beta \ne 0$ give $x^\alpha \cos(\beta \ln x)$ and $x^\alpha \sin(\beta \ln x)$. Example: $x^2 y'' - 2 y = 0$ gives $m(m-1) - 2 = 0$, so $m = 2$ or $m = -1$, and the solutions are $c_1 x^2 + c_2 x^{-1}$.