\[ y''(x)=-\frac {y'(x) \left (-\phi ''(x)-\phi (x) \phi '(x)+\phi \left (x^3\right )\right )}{\phi '(x)+\phi (x)^2}-\frac {y(x) \left (-\phi (x) \phi ''(x)+\phi (x)^2 \left (-\phi '(x)\right )+\phi '(x)^2\right )}{\phi '(x)+\phi (x)^2} \] ✗ Mathematica : cpu = 0.883245 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == -((Derivative[1][y][x]*(phi[x^3] - phi[x]*Derivative[1][phi][x] - Derivative[2][phi][x]))/(phi[x]^2 + Derivative[1][phi][x])) - (y[x]*(-(phi[x]^2*Derivative[1][phi][x]) + Derivative[1][phi][x]^2 - phi[x]*Derivative[2][phi][x]))/(phi[x]^2 + Derivative[1][phi][x]), y[x], x]
✗ Maple : cpu = 0. (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac { \left ( \left ( {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) \right ) ^{2}- \left ( \phi \left ( x \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) -\phi \left ( x \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}\phi \left ( x \right ) \right ) {\it \_Y} \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) + \left ( \phi \left ( x \right ) \right ) ^{2}}}+{\frac { \left ( \phi \left ( {x}^{3} \right ) -\phi \left ( x \right ) {\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) -{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}\phi \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{{\frac {\rm d}{{\rm d}x}}\phi \left ( x \right ) + \left ( \phi \left ( x \right ) \right ) ^{2}}}+{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]