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: 0.312 (sec). Leaf size: 685
dsolve(y(x)+x*(x^2*y(x)-1)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {3+\frac {{\left (\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}-c_{1}^{2}\right )}^{2}}{c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}}}{2 x^{2}} \\ y \left (x \right ) &= \frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}}+c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+c_{1}^{4}}{2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2}} \\ y \left (x \right ) &= \frac {\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}}+c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+c_{1}^{4}}{2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2}} \\ y \left (x \right ) &= \frac {2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}} \left (i \sqrt {3}-1\right )-\left (1+i \sqrt {3}\right ) c_{1}^{4}}{4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2} c_{1}^{2}} \\ y \left (x \right ) &= -\frac {-2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )-c_{1}^{4} \left (i \sqrt {3}-1\right )}{4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2} c_{1}^{2}} \\ y \left (x \right ) &= \frac {2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}} \left (i \sqrt {3}-1\right )-\left (1+i \sqrt {3}\right ) c_{1}^{4}}{4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2} c_{1}^{2}} \\ y \left (x \right ) &= -\frac {-2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )-c_{1}^{4} \left (i \sqrt {3}-1\right )}{4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2} c_{1}^{2}} \\ y \left (x \right ) &= \frac {2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}} \left (i \sqrt {3}-1\right )-\left (1+i \sqrt {3}\right ) c_{1}^{4}}{4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2} c_{1}^{2}} \\ y \left (x \right ) &= -\frac {-2 c_{1}^{2} \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}}+\left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )-c_{1}^{4} \left (i \sqrt {3}-1\right )}{4 \left (x^{3}+\sqrt {c_{1}^{6}+x^{6}}\right )^{\frac {2}{3}} x^{2} c_{1}^{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*}