Internal problem ID [11676]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 2, Miscellaneous Review. Exercises page 60
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _exact, _rational]
\[ \boxed {\left (3 x^{2} y^{2}-x \right ) y^{\prime }+2 y^{3} x -y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 1127
dsolve((3*x^2*y(x)^2-x)*diff(y(x),x)+(2*x*y(x)^3-y(x))=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {{\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} \left (2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}+4 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}\right )}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {{\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} \left (2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}+4 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}\right )}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {\left (i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {\left (i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {\left (-i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {\left (-i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.566 (sec). Leaf size: 356
DSolve[(3*x^2*y[x]^2-x)*y'[x]+(2*x*y[x]^3-y[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4\right ){}^{2/3}}{6^{2/3} x^2 \sqrt [3]{\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4}} \\ y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (2 \sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+18 c_1 x^4\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3}{12 x^2 \sqrt [3]{\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4}} \\ y(x)\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+18 c_1 x^4\right ){}^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3}{12 x^2 \sqrt [3]{\sqrt {3} \sqrt {x^8 \left (-4 x+27 c_1{}^2\right )}+9 c_1 x^4}} \\ \end{align*}