ODE
\[ 2 x^3 y''(x)+x^2 (2 x y(x)+9) y'(x)+x y(x) \left (-2 x^2 y(x)^2+3 x y(x)+12\right )-6=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 25.1892 (sec), leaf count = 0 , could not solve
DSolve[-6 + x*y[x]*(12 + 3*x*y[x] - 2*x^2*y[x]^2) + x^2*(9 + 2*x*y[x])*Derivative[1][y][x] + 2*x^3*Derivative[2][y][x] == 0, y[x], x]
Maple ✓
cpu = 0.809 (sec), leaf count = 96
\[ \left \{ y \left ( x \right ) ={\it ODESolStruc} \left ( {\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}},[ \left \{ {\frac {\rm d}{{\rm d}{\it \_a}}}{\it \_b} \left ( {\it \_a} \right ) =- \left ( {\it \_b} \left ( {\it \_a} \right ) \right ) ^{2} \left ( \left ( {{\it \_a}}^{3}-{\frac {{{\it \_a}}^{2}}{2}}-{\frac {7\,{\it \_a}}{2}}+3 \right ) {\it \_b} \left ( {\it \_a} \right ) +{\it \_a}+{\frac {3}{2}} \right ) \right \} , \left \{ {\it \_a}=xy \left ( x \right ) ,{\it \_b} \left ( {\it \_a} \right ) =-{\frac {1}{x \left ( x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +y \left ( x \right ) \right ) }} \right \} , \left \{ x= \left ( {{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right ) ^{-1},y \left ( x \right ) ={\it \_a}\,{{\rm e}^{\int \!{\it \_b} \left ( {\it \_a} \right ) \,{\rm d}{\it \_a}+{\it \_C1}}} \right \} ] \right ) \right \} \] Mathematica raw input
DSolve[-6 + x*y[x]*(12 + 3*x*y[x] - 2*x^2*y[x]^2) + x^2*(9 + 2*x*y[x])*y'[x] + 2*x^3*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[-6 + x*y[x]*(12 + 3*x*y[x] - 2*x^2*y[x]^2) + x^2*(9 + 2*x*y[x])*Derivative[1][y][x] + 2*x^3*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(2*x^3*diff(diff(y(x),x),x)+x^2*(9+2*x*y(x))*diff(y(x),x)-6+x*y(x)*(12+3*x*y(x)-2*x^2*y(x)^2) = 0, y(x),'implicit')
Maple raw output
y(x) = ODESolStruc(_a*exp(Int(_b(_a),_a)+_C1),[{diff(_b(_a),_a) = -_b(_a)^2*((_a^3-1/2*_a^2-7/2*_a+3)*_b(_a)+_a+3/2)}, {_a = x*y(x), _b(_a) = -1/x/(x*diff(y(x),x)+y(x))}, {x = 1/exp(Int(_b(_a),_a)+_C1), y(x) = _a*exp(Int(_b(_a),_a)+_C1)}])