\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( 1/2\,{\frac {{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}g \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}-3/4\,{\frac { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}g \left ( x \right ) \right ) ^{2}}{ \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2}}}+{\frac { \left ( 1/4-{v}^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2}}{g \left ( x \right ) }}+ \left ( {\frac {\rm d}{{\rm d}x}}g \left ( x \right ) \right ) ^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.478061 (sec), leaf count = 71 \[ \text {DSolve}\left [y(x) \left (\frac {\left (\frac {1}{4}-v^2\right ) g'(x)^2}{g(x)}+g'(x)^2+\frac {g^3(x)}{2 g'(x)}-\frac {3 g''(x)^2}{4 g'(x)^2}\right )+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.156 (sec), leaf count = 53 \[ \left \{ y \left ( x \right ) ={{\it \_C1}{{\sl M}_{{\frac {i}{2}}{v}^{2 }-{\frac {i}{8}},\,{\frac {1}{2}}}\left (2\,ig \left ( x \right ) \right )}{\frac {1}{\sqrt {{\frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}}}+{{\it \_C2}{{\sl W}_{{\frac {i}{2}}{v}^{2}-{\frac {i}{8}},\,{ \frac {1}{2}}}\left (2\,ig \left ( x \right ) \right )}{\frac {1}{\sqrt {{ \frac {\rm d}{{\rm d}x}}g \left ( x \right ) }}}} \right \} \]