\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( {\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}+{\frac { \left ( 2\,v-1 \right ) {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }{g \left ( x \right ) }}+2\,{\frac {{\frac {\rm d}{{\rm d}x}}h \left ( x \right ) }{h \left ( x \right ) }} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( {\frac {{\frac {\rm d}{{\rm d}x}}h \left ( x \right ) }{h \left ( x \right ) } \left ( {\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}+{\frac { \left ( 2\,v-1 \right ) {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }{g \left ( x \right ) }}+2\,{\frac {{\frac {\rm d}{{\rm d}x}}h \left ( x \right ) }{h \left ( x \right ) }} \right ) }-{\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}h \left ( x \right ) }{h \left ( x \right ) }}+ \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.922617 (sec), leaf count = 118 \[ \text {DSolve}\left [-y'(x) \left (\frac {(2 v-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}+\frac {2 h'(x)}{h(x)}\right )+y(x) \left (g'(x)^2+\frac {h'(x) \left (\frac {(2 v-1) g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}+\frac {2 h'(x)}{h(x)}\right )}{h(x)}-\frac {h''(x)}{h(x)}\right )+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.062 (sec), leaf count = 29 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl J}_{v}\left (g \left ( x \right ) \right )}h \left ( x \right ) \left ( g \left ( x \right ) \right ) ^{v}+{\it \_C2}\,{{\sl Y}_{v}\left (g \left ( x \right ) \right )}h \left ( x \right ) \left ( g \left ( x \right ) \right ) ^{v} \right \} \]