Internal problem ID [3176]
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)]`]]
\[ \boxed {y+x \left (y^{2}+\ln \left (x \right )\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 237
dsolve((y(x))+x*(y(x)^2+ln(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}-4 \ln \left (x \right )}{2 \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {i \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+4 i \ln \left (x \right ) \sqrt {3}+\left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}-4 \ln \left (x \right )}{4 \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+4 i \ln \left (x \right ) \sqrt {3}-\left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}+4 \ln \left (x \right )}{4 \left (-12 c_{1} +4 \sqrt {4 \ln \left (x \right )^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.211 (sec). Leaf size: 272
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 {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) \log (x)+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{4 \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \log (x)}{2^{2/3} \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 \log ^3(x)+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}