Internal problem ID [5286]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 5. Equations of first order and first degree (Exact equations). Supplemetary
problems. Page 33
Problem number: 25 (j).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {y+x \left (y x^{2}-1\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 789
dsolve(y(x)+x*(x^2*y(x)-1)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y = \frac {{\left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}-\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )}^{2}+3}{2 x^{2}} y = \frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} {\left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}-\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )}^{2}+3}{2 x^{2}} y = \frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} {\left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}-\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )}^{2}+3}{2 x^{2}} y = \frac {\frac {{\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{2}} y = \frac {\frac {{\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{2}} y = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{2}} y = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{2}} y = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{2}} y = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} {\left (-\frac {4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {4 c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}{c_{1}}+\frac {c_{1}}{\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {1}{3}}}\right )\right )}^{2}}{64}+3}{2 x^{2}} \end{align*}
✓ Solution by Mathematica
Time used: 56.665 (sec). Leaf size: 452
DSolve[y[x]+x*(x^2*y[x]-1)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-6 c_1} \sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}+\frac {e^{6 c_1}}{\sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}}+1}{2 x^2} y(x)\to \frac {i \left (\sqrt {3}+i\right ) e^{-6 c_1} \sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1}}{\sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}}+2}{4 x^2} y(x)\to \frac {-\left (\left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}\right )+\frac {i \left (\sqrt {3}+i\right ) e^{6 c_1}}{\sqrt [3]{-2 e^{12 c_1} x^6+2 \sqrt {-e^{24 c_1} x^6 \left (-x^6+e^{6 c_1}\right )}+e^{18 c_1}}}+2}{4 x^2} y(x)\to 0 y(x)\to \frac {3}{2 x^2} \end{align*}