Internal problem ID [6230]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.8. Integrating Factors. Page
32
Problem number: 1(f).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {y+\left (x -2 y^{3} x^{2}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 432
dsolve(y(x)+(x-2*x^2*y(x)^3)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}{6 c_{1} x}+\frac {2 x}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}} y \left (x \right ) = -\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}{12 c_{1} x}-\frac {x}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}{6 c_{1} x}-\frac {2 x}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}{12 c_{1} x}-\frac {x}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}{6 c_{1} x}-\frac {2 x}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}-108 c_{1} \right ) c_{1}^{2} x^{2}\right )}^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 28.221 (sec). Leaf size: 327
DSolve[y[x]+(x-2*x^2*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt [3]{3} c_1 x^2+\sqrt [3]{2} \left (-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (-18 x^2+2 \sqrt {81 x^4-12 c_1{}^3 x^6}\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) c_1 x^2}{12 x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} y(x)\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (-18 x^2+2 \sqrt {81 x^4-12 c_1{}^3 x^6}\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) c_1 x^2}{12 x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} y(x)\to 0 \end{align*}