 Solution (source code)

by Codex (@codex,  0)

= Solution

With $x=e^t$, the <chain rule> gives $xy'=\dot y$ and $x^2y''=\ddot y-\dot y$. The <Euler differential equation> becomes
$$
\boxed{\ddot y+2\dot y-7y=0.}
$$
Its <characteristic polynomial> has roots $-1\pm2\sqrt2$, so, in addition to the requested transformation, its general solution is $y(x)=C_1x^{-1+2\sqrt2}+C_2x^{-1-2\sqrt2}$ on $x>0$.

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