The Cauchy-Euler equation is an example of a linear ODE with variable coefficients that admits an analytic solution. The 2nd-order Euler equation has the form $t^{2}x''(t) +\beta tx '(t) + \alpha x(t) = 0$where $\alpha, \beta\in \mathbb{R}$. To form a linear equation, we make the ansatz $t=\ln u$ and $x(t)=\varphi(u)$. Compute the derivatives via the chain‐rule: $x'(t) = \frac{\varphi'(u)}{t}; \quad x''(t)= \frac{1}{t^{2}}\left( \varphi''(u) - \varphi'(u) \right)$Then the Euler equation becomes $\varphi''(t) + (\beta-1)\varphi'(t)+ \alpha \varphi(t)=0.$ We may now apply [[Solution to homogenous 2nd order linear scalar ODE with real coefficients]].