4.12.30 \(x (2 x y(x)+1) y'(x)+y(x) (3 x y(x)+2)=0\)

ODE
\[ x (2 x y(x)+1) y'(x)+y(x) (3 x y(x)+2)=0 \] ODE Classification

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

Book solution method
Homogeneous equation, isobaric equation

Mathematica
cpu = 0.0131273 (sec), leaf count = 69

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

Maple
cpu = 0.012 (sec), leaf count = 20

\[ \left \{ \ln \relax (x ) -{\it \_C1}+\ln \left (xy \relax (x ) \left (1+xy \relax (x ) \right ) \right ) =0 \right \} \] Mathematica raw input

DSolve[y[x]*(2 + 3*x*y[x]) + x*(1 + 2*x*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*(1+2*x*y(x))*diff(y(x),x)+(2+3*x*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1+ln(x*y(x)*(1+x*y(x))) = 0