\[ \boxed { {x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +2\,{x}^{2}f \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( {x}^{2} \left ( {\frac {\rm d}{{\rm d}x}}f \left ( x \right ) + \left ( f \left ( x \right ) \right ) ^{2}+a \right ) -v \left ( v-1 \right ) \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 11.805499 (sec), leaf count = 96 \[ \left \{\left \{y(x)\to c_1 J_{\frac {1}{2} (2 v-1)}\left (\sqrt {a} x\right ) \exp \left (\int _1^x \frac {1-2 K[1] f(K[1])}{2 K[1]} \, dK[1]\right )+c_2 Y_{\frac {1}{2} (2 v-1)}\left (\sqrt {a} x\right ) \exp \left (\int _1^x \frac {1-2 K[1] f(K[1])}{2 K[1]} \, dK[1]\right )\right \}\right \} \]
Maple: cpu = 0.015 (sec), leaf count = 51 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {\int \!2\, f \left ( x \right ) \,{\rm d}x}{2}}}}\sqrt {x}{{\sl J}_{v-{\frac {1}{2} }}\left (\sqrt {a}x\right )}+{\it \_C2}\,{{\rm e}^{-{\frac {\int \!2\,f \left ( x \right ) \,{\rm d}x}{2}}}}\sqrt {x}{{\sl Y}_{v-{\frac {1}{2}} }\left (\sqrt {a}x\right )} \right \} \]