\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +a \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}p \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( a+bp \left ( x \right ) -4\,na \left ( p \left ( x \right ) \right ) ^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.281536 (sec), leaf count = 38 \[ \text {DSolve}\left [y(x) \left (-4 a n p(x)^2+a+b p(x)\right )+a p''(x) y'(x)+y''(x)=0,y(x),x\right ] \]
Maple: cpu = 0.250 (sec), leaf count = 45 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac { {\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +a \left ( { \frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}p \left ( x \right ) \right ) { \frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) + \left ( a+bp \left ( x \right ) -4\,na \left ( p \left ( x \right ) \right ) ^{2} \right ) {\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]