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

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

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.0305008 (sec), leaf count = 66

\[\text {Solve}\left [c_1=\text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}+1\& ,\frac {3 \text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{3 \text {$\#$1}^2+1}\& \right ]+3 \log (x),y(x)\right ]\]

Maple
cpu = 0.014 (sec), leaf count = 38

\[ \left \{ -{\frac {1}{3}\int ^{{\frac {y \relax (x ) }{x}}}\!{\frac {3\,{{\it \_a}}^{2}+2}{{{\it \_a}}^{3}+{\it \_a}+1}}{d{\it \_a}}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[C[1] == 3*Log[x] + RootSum[1 + #1 + #1^3 & , (2*Log[-#1 + y[x]/x] + 3*Log[
-#1 + y[x]/x]*#1^2)/(1 + 3*#1^2) & ], y[x]]

Maple raw input

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

Maple raw output

-1/3*Intat(1/(_a^3+_a+1)*(3*_a^2+2),_a = y(x)/x)-ln(x)-_C1 = 0