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

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

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

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.0106324 (sec), leaf count = 73

\[\left \{\left \{y(x)\to 2 x^3 \left (\frac {2}{1-\frac {4 x^{3/2}}{\sqrt {c_1+16 x^3}}}-1\right )\right \},\left \{y(x)\to 2 x^3 \left (\frac {2}{\frac {4 x^{3/2}}{\sqrt {c_1+16 x^3}}+1}-1\right )\right \}\right \}\]

Maple
cpu = 0.02 (sec), leaf count = 33

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

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

Mathematica raw output

{{y[x] -> 2*x^3*(-1 + 2/(1 - (4*x^(3/2))/Sqrt[16*x^3 + C[1]]))}, {y[x] -> 2*x^3*
(-1 + 2/(1 + (4*x^(3/2))/Sqrt[16*x^3 + C[1]]))}}

Maple raw input

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

Maple raw output

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