\[ y(x) \left (\frac {\left (m^2-v^2\right ) g'(x)^2}{g(x)}+g'(x)^2\right )-y'(x) \left (\frac {(2 m-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.693124 (sec), leaf count = 0 , could not solve
DSolve[y[x]*(Derivative[1][g][x]^2 + ((m^2 - v^2)*Derivative[1][g][x]^2)/g[x]) - Derivative[1][y][x]*(((-1 + 2*m)*Derivative[1][g][x])/g[x] + Derivative[2][g][x]/Derivative[1][g][x]) + Derivative[2][y][x] == 0, y[x], x]
✓ Maple : cpu = 0.153 (sec), leaf count = 74
\[ \left \{ y \left ( x \right ) = \left ( g \left ( x \right ) \right ) ^{2\,m}{{\rm e}^{-ig \left ( x \right ) }} \left ( {{\sl U}\left ({\frac {i}{2}}{m}^{2}-{\frac {i}{2}}{v}^{2}+m+{\frac {1}{2}},\,2\,m+1,\,2\,ig \left ( x \right ) \right )}{\it \_C2}+{{\sl M}\left ({\frac {i}{2}}{m}^{2}-{\frac {i}{2}}{v}^{2}+m+{\frac {1}{2}},\,2\,m+1,\,2\,ig \left ( x \right ) \right )}{\it \_C1} \right ) \right \} \]