\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-{\frac { \left ( \phi \left ( {x}^{3} \right ) -\phi \left ( x \right ) {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) -{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}\phi \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) + \left ( \phi \left ( x \right ) \right ) ^{2}}}-{\frac { \left ( \left ( {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) \right ) ^{2}- \left ( \phi \left ( x \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) -\phi \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}\phi \left ( x \right ) \right ) y \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) + \left ( \phi \left ( x \right ) \right ) ^{2}}}=0} \]
Mathematica: cpu = 0.890613 (sec), leaf count = 86 \[ \text {DSolve}\left [y''(x)=-\frac {y'(x) \left (-\phi ''(x)-\phi (x) \phi '(x)+\phi \left (x^3\right )\right )}{\phi '(x)+\phi (x)^2}-\frac {y(x) \left (-\phi (x) \phi ''(x)+\phi (x)^2 \left (-\phi '(x)\right )+\phi '(x)^2\right )}{\phi '(x)+\phi (x)^2},y(x),x\right ] \]
Maple: cpu = 0.0 (sec), leaf count = 94 \[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac { \left ( \left ( {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) \right ) ^{2}- \left ( \phi \left ( x \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) -\phi \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}\phi \left ( x \right ) \right ) {\it \_Y} \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) + \left ( \phi \left ( x \right ) \right ) ^{2}}}+{\frac { \left ( \phi \left ( {x}^{3} \right ) -\phi \left ( x \right ) {\frac {\rm d}{{\rm d}x }}\phi \left ( x \right ) -{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}\phi \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) + \left ( \phi \left ( x \right ) \right ) ^{2}}}+{\frac {{\rm d}^{2}}{{\rm d}{x}^ {2}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]