Exercise 38

An important class of second-order linear differential equations are equations of the form $$t^2 y'' + \alpha t y' + \beta y = 0,$$ where $t \gt 0$ and $\alpha$ and $\beta$ are real constants. An equation of this form is called an Euler equation.

1. Show that the substitution $x = \ln t$ transforms an Euler equation into an equation of constant coefficients.

2. Solve the equation $$t^2 y'' + 4ty' + 2y = 0.$$