ODE
\[ x y(x) y''(x)=a y(x) y'(x)+x y'(x)^2+x y(x)^3 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 51.8477 (sec), leaf count = 0 , could not solve
DSolve[x*y[x]*Derivative[2][y][x] == x*y[x]^3 + a*y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2, y[x], x]
Maple ✓
cpu = 1.152 (sec), leaf count = 107
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\frac {{\it \_a}}{ \left ( {{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right ) ^{2}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =2\,{\frac {{\it \_b} \left ( {\it \_a} \right ) \left ( -1/2+{{\it \_a}}^{2} \left ( a-{\it \_a}/2+1 \right ) \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}-1/2\,{\it \_a}\, \left ( 1+a \right ) {\it \_b} \left ( {\it \_a} \right ) \right ) }{{\it \_a}}} \right \} , \left \{ {\it \_a}={x}^{2}y \left ( x \right ) ,{\it \_b} \left ( {\it \_a} \right ) ={\frac {1}{{x}^{2} \left ( x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +2\,y \left ( x \right ) \right ) }} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={\frac {{\it \_a}}{ \left ( {{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right ) ^{2}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[x*y[x]*y''[x] == x*y[x]^3 + a*y[x]*y'[x] + x*y'[x]^2,y[x],x]
Mathematica raw output
DSolve[x*y[x]*Derivative[2][y][x] == x*y[x]^3 + a*y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2, y[x], x]
Maple raw input
dsolve(x*y(x)*diff(diff(y(x),x),x) = x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+x*y(x)^3, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a/exp(Int(_b(_a),_a)+_C1)^2,[{diff(_b(_a),_a) = 2*_b(_a)*(-1/2+_a^2*(a-1/2*_a+1)*_b(_a)^2-1/2*_a*(1+a)*_b(_a))/_a}, {_a = x^2*y(x), _b(_a) = 1/x^2/(x*diff(y(x),x)+2*y(x))}, {x = exp(Int(_b(_a),_a)+_C1), y(x) = _a/exp(Int(_b(_a),_a)+_C1)^2}])