\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( {\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}+{\frac { \left ( 2\,m-1 \right ) {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }{g \left ( x \right ) }} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( {\frac { \left ( {m}^{2}-{v}^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2}}{g \left ( x \right ) }}+ \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.548070 (sec), leaf count = 73 \[ \text {DSolve}\left [y(x) \left (\frac {\left (m^2-v^2\right ) g'(x)^2}{g(x)}+g'(x)^2\right )-y'(x) \left (\frac {(2 m-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.094 (sec), leaf count = 85 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{-ig \left ( x \right ) }}{{\sl M}\left ({\frac {i}{2}}{m}^{2}-{\frac {i}{2}}{v}^{2}+m +{\frac {1}{2}},\,2\,m+1,\,2\,ig \left ( x \right ) \right )} \left ( g \left ( x \right ) \right ) ^{2\,m}+{\it \_C2}\,{{\rm e}^{-ig \left ( x \right ) }}{{\sl U}\left ({\frac {i}{2}}{m}^{2}-{\frac {i}{2}}{v}^{2}+m +{\frac {1}{2}},\,2\,m+1,\,2\,ig \left ( x \right ) \right )} \left ( g \left ( x \right ) \right ) ^{2\,m} \right \} \]