4.12.10 \(x^3+3 x (2 y(x)+x) y'(x)+3 y(x) (y(x)+2 x)=0\)

ODE
\[ x^3+3 x (2 y(x)+x) y'(x)+3 y(x) (y(x)+2 x)=0 \] ODE Classification

[_exact, _rational, [_Abel, `2nd type``class B`]]

Book solution method
Exact equation

Mathematica
cpu = 0.0110045 (sec), leaf count = 75

\[\left \{\left \{y(x)\to -\frac {\sqrt {36 c_1 x-3 x^5+9 x^4}+3 x^2}{6 x}\right \},\left \{y(x)\to \frac {\sqrt {36 c_1 x-3 x^5+9 x^4}-3 x^2}{6 x}\right \}\right \}\]

Maple
cpu = 0.014 (sec), leaf count = 24

\[ \left \{ {\frac {{x}^{4}}{4}}+3\,{x}^{2}y \relax (x ) +3\,x \left (y \relax (x ) \right ) ^{2}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x^3 + 3*y[x]*(2*x + y[x]) + 3*x*(x + 2*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(3*x^2 + Sqrt[9*x^4 - 3*x^5 + 36*x*C[1]])/(6*x)}, {y[x] -> (-3*x^2 + 
Sqrt[9*x^4 - 3*x^5 + 36*x*C[1]])/(6*x)}}

Maple raw input

dsolve(3*x*(x+2*y(x))*diff(y(x),x)+x^3+3*y(x)*(2*x+y(x)) = 0, y(x),'implicit')

Maple raw output

1/4*x^4+3*x^2*y(x)+3*x*y(x)^2+_C1 = 0