4.11.38 \(x \left (2 x^3+y(x)\right ) y'(x)=\left (2 x^3-y(x)\right ) y(x)\)

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

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0111322 (sec), leaf count = 71

\[\left \{\left \{y(x)\to \frac {2 x^4}{\frac {\sqrt {4 c_1 x^4+1}}{\sqrt {\frac {1}{x^2}}}-x}\right \},\left \{y(x)\to -\frac {2 x^4}{\frac {\sqrt {4 c_1 x^4+1}}{\sqrt {\frac {1}{x^2}}}+x}\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 31

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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