Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 10 / a

With $x=e^t$, the chain rule gives $x{y}^{\prime }=\dot{y}$ and $x^2{y}^{\prime \prime }=\ddot{y}-\dot{y}$. The Euler differential equation becomesIts characteristic polynomial has roots $-1\pm 2\sqrt{2}$, so, in addition to the requested transformation, its general solution is $y(x)=C_1{x}^{-1+2\sqrt{2}}+C_2{x}^{-1-2\sqrt{2}}$ on $x>0$.