\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) ={\frac { \left ( {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{\phi \left ( x \right ) -\phi \left ( a \right ) }}-{\frac { \left ( -n \left ( n+1 \right ) \left ( \phi \left ( x \right ) -\phi \left ( a \right ) \right ) ^{2}+ \left ( D^{ \left ( 2 \right ) } \right ) \left ( \phi \right ) \left ( a \right ) \right ) y \left ( x \right ) }{\phi \left ( x \right ) -\phi \left ( a \right ) }}=0} \]
Mathematica: cpu = 0.765097 (sec), leaf count = 61 \[ \text {DSolve}\left [y''(x)=\frac {\phi '(x) y'(x)}{\phi (x)-\phi (a)}-\frac {y(x) \left (\phi ''(a)-n (n+1) (\phi (x)-\phi (a))^2\right )}{\phi (x)-\phi (a)},y(x),x\right ] \]
Maple: cpu = 0.671 (sec), leaf count = 69 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac { {\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) -{\frac { \left ( {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) \right ) { \frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{\phi \left ( x \right ) -\phi \left ( a \right ) }}+{\frac { \left ( -n \left ( n+1 \right ) \left ( \phi \left ( x \right ) -\phi \left ( a \right ) \right ) ^{2}+{\frac {{\rm d}^{2}}{{\rm d}{a}^{2}}}\phi \left ( a \right ) \right ) {\it \_Y} \left ( x \right ) }{\phi \left ( x \right ) - \phi \left ( a \right ) }} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]