Internal problem ID [11702]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 2, Section 2.4. Special integrating factors and transformations. Exercises page
67
Problem number: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {2 y^{2} x +y+\left (2 y^{3}-x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 301
dsolve((2*x*y(x)^2+y(x))+(2*y(x)^3-x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {-12 x^{2}-12 c_{1} +\left (-108 x +12 \sqrt {12 x^{6}+36 x^{4} c_{1} +\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{\frac {2}{3}}}{6 \left (-108 x +12 \sqrt {12 x^{6}+36 x^{4} c_{1} +\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108 x +12 \sqrt {12 x^{6}+36 x^{4} c_{1} +\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{\frac {2}{3}}+\left (x^{2}+c_{1} \right ) \left (i \sqrt {3}-1\right )}{\left (-108 x +12 \sqrt {12 x^{6}+36 x^{4} c_{1} +\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-108 x +12 \sqrt {12 x^{6}+36 x^{4} c_{1} +\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{\frac {2}{3}}}{12}+\left (x^{2}+c_{1} \right ) \left (1+i \sqrt {3}\right )}{\left (-108 x +12 \sqrt {12 x^{6}+36 x^{4} c_{1} +\left (36 c_{1}^{2}+81\right ) x^{2}+12 c_{1}^{3}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.163 (sec). Leaf size: 316
DSolve[(2*x*y[x]^2+y[x])+(2*y[x]^3-x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{2/3} \left (-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}\right ){}^{2/3}-6 \sqrt [3]{2} \left (x^2-c_1\right )}{6 \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^2-c_1\right )}{2^{2/3} \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}{6 \sqrt [3]{2}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^2-c_1\right )}{2^{2/3} \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}{6 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}