ODE
\[ y'(x) y''(x)=a x y'(x)^5+3 y''(x)^2 \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✗
cpu = 9.60746 (sec), leaf count = 0 , could not solve
DSolve[Derivative[1][y][x]*Derivative[2][y][x] == a*x*Derivative[1][y][x]^5 + 3*Derivative[2][y][x]^2, y[x], x]
Maple ✓
cpu = 0.726 (sec), leaf count = 60
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( \int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =-{\frac {{\it \_b} \left ( {\it \_a} \right ) }{6} \left ( -1+\sqrt {1-12\,a{\it \_a}\, \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{3}} \right ) } \right \} , \left \{ {\it \_a}=x,{\it \_b} \left ( {\it \_a} \right ) ={\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right \} , \left \{ x={\it \_a},y \left ( x \right ) =\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[y'[x]*y''[x] == a*x*y'[x]^5 + 3*y''[x]^2,y[x],x]
Mathematica raw output
DSolve[Derivative[1][y][x]*Derivative[2][y][x] == a*x*Derivative[1][y][x]^5 + 3*Derivative[2][y][x]^2, y[x], x]
Maple raw input
dsolve(diff(y(x),x)*diff(diff(y(x),x),x) = a*x*diff(y(x),x)^5+3*diff(diff(y(x),x),x)^2, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(Int(_b(_a),_a)+_C1,[{diff(_b(_a),_a) = -1/6*(-1+(1-12*a*_a*_b(_a)^3)^(1/2))*_b(_a)}, {_a = x, _b(_a) = diff(y(x),x)}, {x = _a, y(x) = Int(_b(_a),_a)+_C1}])