ODE
\[ a x y'(x)+y(x) y''(x)+y'(x)^2=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 46.9385 (sec), leaf count = 0 , could not solve
DSolve[a*x*Derivative[1][y][x] + Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 0.541 (sec), leaf count = 110
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a}\, \left ( {{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right ) ^{2},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) ={\frac {{\it \_b} \left ( {\it \_a} \right ) \left ( 6\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{{\it \_a}}^{2}+2\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2}{\it \_a}\,a+7\,{\it \_b} \left ( {\it \_a} \right ) {\it \_a}+{\it \_b} \left ( {\it \_a} \right ) a+1 \right ) }{{\it \_a}}} \right \} , \left \{ {\it \_a}={\frac {y \left ( x \right ) }{{x}^{2}}},{\it \_b} \left ( {\it \_a} \right ) ={\frac {{x}^{2}}{x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -2\,y \left ( x \right ) }} \right \} , \left \{ x={{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},y \left ( x \right ) ={\it \_a}\, \left ( {{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right ) ^{2} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[a*x*y'[x] + y'[x]^2 + y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[a*x*Derivative[1][y][x] + Derivative[1][y][x]^2 + y[x]*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2+a*x*diff(y(x),x) = 0, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a*exp(Int(_b(_a),_a)+_C1)^2,[{diff(_b(_a),_a) = _b(_a)*(6*_b(_a)^2*_a^2+2*_b(_a)^2*_a*a+7*_b(_a)*_a+_b(_a)*a+1)/_a}, {_a = 1/x^2*y(x), _b(_a) = 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}])