solving differntial equation through transformation

Question

Transform the following differential equation into a second order differential equation such that the dependent variable is missing. Solve the corresponding differential equation. $$ x^2 y^{''}-3xy'+4y=x^{1/2} $$

Answer 1

HINT

I would say after substitution

$\ln x = t\\ y(x) = z(\ln x)=z(t)\\ \displaystyle \frac{dy}{dx}=\frac{dz}{dt}\cdot\frac{dt}{dx}=\frac1x\cdot\frac{dz}{dt},\quad \frac{d^2y}{dx^2}=\cdots=\frac1{x^2}\left(\frac{d^2z}{dt^2}-\frac{dz}{dt}\right)$

we get an equation with constant coefficients: $\displaystyle \quad z''-4z'+4z=e^{t/2}$