Euler Differential Equations

Euler differential equations are of the type

$$a{x}^2{y}^{\prime \prime }+bx{y}^{\prime }+cy=0$$

introducing the new variable and function $z=lnx$ and $Y(z)=y(x)$ the differential equation can be simplified to

$$aY(z{)}^{\prime \prime }+(b-a)Y(z{)}^{\prime }+cY(z)=0$$

and then solved with the Auxiliary Equation. The resulting solution will still be in the form $Y(z)=...$ . Therefore re-substitution of $x$ into the solution is necessary.