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: 0.125 (sec). Leaf size: 320
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 {12^{\frac {1}{3}} \left (12^{\frac {1}{3}} c_{1} x^{2}+\left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{\frac {2}{3}}\right )}{6 c_{1} x \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (\left (-i \sqrt {3}-1\right ) \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{\frac {2}{3}}+2^{\frac {2}{3}} x^{2} c_{1} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )\right ) 3^{\frac {1}{3}}}{12 \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{\frac {1}{3}} x c_{1}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (\left (1-i \sqrt {3}\right ) \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{\frac {2}{3}}+2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x^{2} c_{1} \right ) 3^{\frac {1}{3}}}{12 \left (-9 x^{2} \left (-\frac {\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3}-4 x^{2}}{c_{1}}}}{9}+c_{1} \right ) c_{1}^{2}\right )^{\frac {1}{3}} x c_{1}} \\ \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*}