Internal problem ID [5443]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page
20
Problem number: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {x \ln \relax (y)+y x +\left (y \ln \relax (x )+y x \right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 28
dsolve((x*ln(y(x))+x*y(x))+(y(x)*ln(x)+x*y(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ \int \frac {x}{x +\ln \relax (x )}d x +\int _{}^{y \relax (x )}\frac {\textit {\_a}}{\ln \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 84.657 (sec). Leaf size: 48
DSolve[(x*Log[y[x]]+x*y[x])+(y[x]*Log[x]+x*y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]}{K[1]+\log (K[1])}dK[1]\&\right ]\left [\int _1^x-\frac {K[2]}{K[2]+\log (K[2])}dK[2]+c_1\right ] \\ \end{align*}