\[ y(x) \left (\frac {\left (\frac {1}{4}-v^2\right ) g'(x)^2}{g(x)}-\frac {3 g''(x)^2}{4 g'(x)^2}+g'(x)^2+\frac {g^3(x)}{2 g'(x)}\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.407629 (sec), leaf count = 0 , could not solve
DSolve[y[x]*((g^3)[x]/(2*Derivative[1][g][x]) + Derivative[1][g][x]^2 + ((1/4 - v^2)*Derivative[1][g][x]^2)/g[x] - (3*Derivative[2][g][x]^2)/(4*Derivative[1][g][x]^2)) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.252 (sec), leaf count = 48
\[ \left \{ y \left ( x \right ) ={ \left ( {\it \_C2}\,{{\sl W}_{{\frac {i}{2}}{v}^{2}-{\frac {i}{8}},\,{\frac {1}{2}}}\left (2\,ig \left ( x \right ) \right )}+{\it \_C1}\,{{\sl M}_{{\frac {i}{2}}{v}^{2}-{\frac {i}{8}},\,{\frac {1}{2}}}\left (2\,ig \left ( x \right ) \right )} \right ) {\frac {1}{\sqrt {{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}}} \right \} \]