4.15.3 \(y(x) \left (1-2 x^3 y(x)\right )+x \left (1-2 x y(x)^3\right ) y'(x)=0\)

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

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0305952 (sec), leaf count = 358

\[\left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (c_1 x-x^3\right )}{\sqrt [3]{\sqrt {108 x^3 \left (x^3-c_1 x\right ){}^3+729 x^4}-27 x^2}}+\frac {\sqrt [3]{\sqrt {108 x^3 \left (x^3-c_1 x\right ){}^3+729 x^4}-27 x^2}}{3 \sqrt [3]{2} x}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{\sqrt {108 x^3 \left (x^3-c_1 x\right ){}^3+729 x^4}-27 x^2}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {108 x^3 \left (x^3-c_1 x\right ){}^3+729 x^4}-27 x^2}}{6 \sqrt [3]{2} x}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{\sqrt {108 x^3 \left (x^3-c_1 x\right ){}^3+729 x^4}-27 x^2}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {108 x^3 \left (x^3-c_1 x\right ){}^3+729 x^4}-27 x^2}}{6 \sqrt [3]{2} x}\right \}\right \}\]

Maple
cpu = 0.108 (sec), leaf count = 20

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

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

Mathematica raw output

{{y[x] -> (2^(1/3)*(-x^3 + x*C[1]))/(-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C
[1])^3])^(1/3) + (-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C[1])^3])^(1/3)/(3*2
^(1/3)*x)}, {y[x] -> ((1 + I*Sqrt[3])*(x^3 - x*C[1]))/(2^(2/3)*(-27*x^2 + Sqrt[7
29*x^4 + 108*x^3*(x^3 - x*C[1])^3])^(1/3)) - ((1 - I*Sqrt[3])*(-27*x^2 + Sqrt[72
9*x^4 + 108*x^3*(x^3 - x*C[1])^3])^(1/3))/(6*2^(1/3)*x)}, {y[x] -> ((1 - I*Sqrt[
3])*(x^3 - x*C[1]))/(2^(2/3)*(-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C[1])^3]
)^(1/3)) - ((1 + I*Sqrt[3])*(-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C[1])^3])
^(1/3))/(6*2^(1/3)*x)}}

Maple raw input

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

Maple raw output

_C1+x^2+1/x/y(x)+y(x)^2 = 0