\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-1/2\,{\frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{f \left ( x \right ) }}-{\frac {g \left ( x \right ) y \left ( x \right ) }{f \left ( x \right ) }}=0} \]
Mathematica: cpu = 0.266534 (sec), leaf count = 36 \[ \text {DSolve}\left [y''(x)=-\frac {f'(x) y'(x)}{2 f(x)}-\frac {g(x) y(x)}{f(x)},y(x),x\right ] \]
Maple: cpu = 0.016 (sec), leaf count = 39 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {g \left ( x \right ) {\it \_Y} \left ( x \right ) }{f \left ( x \right ) }}+{ \frac { \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) { \frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{2\,f \left ( x \right ) }}+{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]