4.12.9 \(\left (x^2+6 x y(x)+3\right ) y'(x)+3 y(x)^2+2 x y(x)+2 x=0\)

ODE
\[ \left (x^2+6 x y(x)+3\right ) y'(x)+3 y(x)^2+2 x y(x)+2 x=0 \] ODE Classification

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

Book solution method
Exact equation

Mathematica
cpu = 0.011937 (sec), leaf count = 83

\[\left \{\left \{y(x)\to -\frac {\sqrt {36 c_1 x+x^4-12 x^3+6 x^2+9}+x^2+3}{6 x}\right \},\left \{y(x)\to -\frac {-\sqrt {36 c_1 x+x^4-12 x^3+6 x^2+9}+x^2+3}{6 x}\right \}\right \}\]

Maple
cpu = 0.014 (sec), leaf count = 25

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

DSolve[2*x + 2*x*y[x] + 3*y[x]^2 + (3 + x^2 + 6*x*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(3 + x^2 + Sqrt[9 + 6*x^2 - 12*x^3 + x^4 + 36*x*C[1]])/(6*x)}, {y[x] 
-> -(3 + x^2 - Sqrt[9 + 6*x^2 - 12*x^3 + x^4 + 36*x*C[1]])/(6*x)}}

Maple raw input

dsolve((3+6*x*y(x)+x^2)*diff(y(x),x)+2*x+2*x*y(x)+3*y(x)^2 = 0, y(x),'implicit')

Maple raw output

x^2*y(x)+3*x*y(x)^2+x^2+_C1+3*y(x) = 0