\[ y''(x)=-\frac {\left (e^{2/x}-v^2\right ) y(x)}{x^4} \] ✓ Mathematica : cpu = 0.442009 (sec), leaf count = 173
DSolve[Derivative[2][y][x] == -(((E^(2/x) - v^2)*y[x])/x^4),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_1 2^{v+\frac {v+1}{2}} \left (e^{2/x}\right )^{\frac {v+1}{2}-\frac {1}{2}} \left (-e^{2/x}\right )^{\frac {1}{2} (-v-1)+\frac {1}{2}} I_v\left (\sqrt {-e^{2/x}}\right )}{\log \left (e^{2/x}\right )}+\frac {c_2 (-1)^{-v} 2^{v+\frac {v+1}{2}} \left (e^{2/x}\right )^{\frac {v+1}{2}-\frac {1}{2}} \left (-e^{2/x}\right )^{\frac {1}{2} (-v-1)+\frac {1}{2}} K_v\left (\sqrt {-e^{2/x}}\right )}{\log \left (e^{2/x}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.06 (sec), leaf count = 23
dsolve(diff(diff(y(x),x),x) = -(exp(2/x)-v^2)/x^4*y(x),y(x))
\[y \left (x \right ) = x \left (\BesselY \left (v , {\mathrm e}^{\frac {1}{x}}\right ) c_{2}+\BesselJ \left (v , {\mathrm e}^{\frac {1}{x}}\right ) c_{1}\right )\]