\[ a e^{-2 x} y(x)+y''(x)+y'(x)=0 \] ✓ Mathematica : cpu = 0.0124364 (sec), leaf count = 37
\[\left \{\left \{y(x)\to c_1 \cos \left (\sqrt {a} e^{-x}\right )-c_2 \sin \left (\sqrt {a} e^{-x}\right )\right \}\right \}\] ✓ Maple : cpu = 0.017 (sec), leaf count = 27
\[y \relax (x ) = c_{1} \sin \left ({\mathrm e}^{-x} \sqrt {a}\right )+c_{2} \cos \left ({\mathrm e}^{-x} \sqrt {a}\right )\]
Hand solution
\[ y^{\prime \prime }+y^{\prime }+ae^{-2x}y=0 \]
Let \(y\relax (x) =\eta \left (\xi \right ) \) where \(\xi =e^{-x}\), hence
\begin {align*} \frac {dy}{dx} & =\frac {d\eta }{d\xi }\frac {d\xi }{dx}\\ & =\frac {d\eta }{d\xi }\left (-e^{-x}\right ) \end {align*}
And
\begin {align*} \frac {d^{2}y}{dx^{2}} & =\frac {d}{dx}\left (\frac {d\eta }{d\xi }\left ( -e^{-x}\right ) \right ) \\ & =\frac {d^{2}\eta }{d\xi ^{2}}\frac {d\xi }{dx}\left (-e^{-x}\right ) +\frac {d\eta }{d\xi }\left (e^{-x}\right ) \\ & =\frac {d^{2}\eta }{d\xi ^{2}}\left (-e^{-x}\right ) \left (-e^{-x}\right ) +\frac {d\eta }{d\xi }\left (e^{-x}\right ) \\ & =\frac {d^{2}\eta }{d\xi ^{2}}\left (e^{-2x}\right ) +\frac {d\eta }{d\xi }\left (e^{-x}\right ) \end {align*}
Hence the original ODE becomes
\begin {align*} \frac {d^{2}\eta }{d\xi ^{2}}\left (e^{-2x}\right ) +\frac {d\eta }{d\xi }\left ( e^{-x}\right ) +\frac {d\eta }{d\xi }\left (-e^{-x}\right ) +ae^{-2x}\eta \left ( \xi \right ) & =0\\ \eta ^{\prime \prime }+a\eta & =0 \end {align*}
This is standard second order with constant coefficients. The solution is
\[ \eta =c_{1}\cos \left (\sqrt {a}\xi \right ) +c_{2}\sin \left (\sqrt {a}\xi \right ) \]
Substituting back
\[ y\relax (x) =c_{1}\cos \left (\sqrt {a}e^{-x}\right ) +c_{2}\sin \left ( \sqrt {a}e^{-x}\right ) \]
Verification