Internal problem ID [2667]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 31.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y+x \left (\ln \relax (x )+y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 275
dsolve((y(x))+x*(y(x)^2+ln(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 \ln \relax (x )}{\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {\ln \relax (x )}{\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 \ln \relax (x )}{\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {\ln \relax (x )}{\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 \ln \relax (x )}{\left (-12 c_{1}+4 \sqrt {4 \ln \relax (x )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.278 (sec). Leaf size: 233
DSolve[(y[x])+x*(y[x]^2+Log[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} \log (x)}{\sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {2 \sqrt [3]{-2} \log (x)+(-2)^{2/3} \left (\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2 \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}} \\ y(x)\to -\frac {2 (-1)^{2/3} \log (x)+\sqrt [3]{-2} \left (\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}} \\ y(x)\to 0 \\ \end{align*}