4.15.3 $y(x)(1-2x^{3}y(x))+x(1-2xy(x{)}^3)y^{\prime }(x)=0$

ODE
$$y(x)(1-2x^{3}y(x))+x(1-2xy(x{)}^3)y^{\prime }(x)=0$$ ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

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

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

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

$$\{\mathit{\_}\mathit{C}\mathit{1}+x^2+\frac{1}{xy\left(x\right)}+{\left(y\left(x\right)\right)}^2=0\}$$ 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