\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{f \left ( x \right ) }}+ \left ( 3/4\,{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) ^{2}}{ \left ( f \left ( x \right ) \right ) ^{2}}}-1/2\,{\frac {{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left ( x \right ) }{f \left ( x \right ) }}-3/4\,{\frac { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left ( x \right ) \right ) ^{2}}{ \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2}}}+1/2\,{\frac {{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}+{\frac { \left ( 1/4-{v}^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2}}{ \left ( g \left ( x \right ) \right ) ^{2}}}+ \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.823605 (sec), leaf count = 111 \[ \text {DSolve}\left [-\frac {f'(x) y'(x)}{f(x)}+y(x) \left (-\frac {f''(x)}{2 f(x)}+\frac {3 f'(x)^2}{4 f(x)^2}+\frac {\left (\frac {1}{4}-v^2\right ) g'(x)^2}{g(x)^2}+g'(x)^2+\frac {g^3(x)}{2 g'(x)}-\frac {3 g''(x)^2}{4 g'(x)^2}\right )+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.078 (sec), leaf count = 43 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sqrt {{\frac {f \left ( x \right ) g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}}{{\sl J}_{v}\left (g \left ( x \right ) \right )}+{\it \_C2}\, \sqrt {{\frac {f \left ( x \right ) g \left ( x \right ) }{{\frac {\rm d}{ {\rm d}x}}g \left ( x \right ) }}}{{\sl Y}_{v}\left (g \left ( x \right ) \right )} \right \} \]