Internal problem ID [5477]
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]
Solve \begin {gather*} \boxed {y+\left (x -2 x^{2} y^{3}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.105 (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 \relax (x ) = \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 \relax (x ) = -\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 \relax (x ) = -\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: 5.565 (sec). Leaf size: 287
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 {\sqrt [3]{-1} \left (\sqrt [3]{-2} \left (-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}\right ){}^{2/3}-2 \sqrt [3]{3} c_1 x^2\right )}{6^{2/3} x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} \\ y(x)\to \frac {\sqrt [3]{-1} \left (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}\right )}{6^{2/3} x \sqrt [3]{-9 x^2+\sqrt {81 x^4-12 c_1{}^3 x^6}}} \\ y(x)\to 0 \\ \end{align*}