\[ y(x) \left (\frac {\left (m^2-v^2\right ) g'(x)^2}{g(x)}+g'(x)^2\right )-y'(x) \left (\frac {(2 m-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y''(x)=0 \] ✗ Mathematica : cpu = 0.539864 (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 = 0.152 (sec), leaf count = 85
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{-ig \left ( x \right ) }}{{\sl M}\left ({\frac {i}{2}}{m}^{2}-{\frac {i}{2}}{v}^{2}+m+{\frac {1}{2}},\,1+2\,m,\,2\,ig \left ( x \right ) \right )} \left ( g \left ( x \right ) \right ) ^{2\,m}+{\it \_C2}\,{{\rm e}^{-ig \left ( x \right ) }}{{\sl U}\left ({\frac {i}{2}}{m}^{2}-{\frac {i}{2}}{v}^{2}+m+{\frac {1}{2}},\,1+2\,m,\,2\,ig \left ( x \right ) \right )} \left ( g \left ( x \right ) \right ) ^{2\,m} \right \} \]