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

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

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type``class B`]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0279902 (sec), leaf count = 71

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

Maple
cpu = 0.013 (sec), leaf count = 35

\[ \left \{ -{\frac {1}{4}\ln \left ({\frac {3\,{x}^{2}+8\,xy \relax (x ) +6\, \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

-1/4*ln((3*x^2+8*x*y(x)+6*y(x)^2)/x^2)-ln(x)-_C1 = 0