\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( 2\,{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{f \left ( x \right ) }}+{\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}-{\frac {{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }{g \left ( x \right ) }} \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( {\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{f \left ( x \right ) } \left ( 2\,{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{f \left ( x \right ) }}+{\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}-{\frac {{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }{g \left ( x \right ) }} \right ) }-{\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left ( x \right ) }{f \left ( x \right ) }}-{\frac {{v}^{2} \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2}}{ \left ( g \left ( x \right ) \right ) ^{2}}}+ \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.859609 (sec), leaf count = 125 \[ \text {DSolve}\left [-y'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )+y(x) \left (-\frac {f''(x)}{f(x)}+\frac {f'(x) \left (\frac {2 f'(x)}{f(x)}-\frac {g'(x)}{g(x)}+\frac {g''(x)}{g'(x)}\right )}{f(x)}-\frac {v^2 g'(x)^2}{g(x)^2}+g'(x)^2\right )+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.078 (sec), leaf count = 21 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\sl J}_{v}\left (g \left ( x \right ) \right )}f \left ( x \right ) +{\it \_C2}\,{{\sl Y}_{v}\left (g \left ( x \right ) \right )}f \left ( x \right ) \right \} \]