76.4.11 problem 11

Internal problem ID [17326]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 11
Date solved : Thursday, March 13, 2025 at 09:27:17 AM
CAS classification : [_separable]

\begin{align*} x \ln \left (y\right )+x y+\left (y \ln \left (x \right )+x y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 28

ode:=x*ln(y(x))+x*y(x)+(y(x)*ln(x)+x*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \int \frac {x}{x +\ln \left (x \right )}d x +\int _{}^{y}\frac {\textit {\_a}}{\ln \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} +c_{1} = 0 \]

✓ Mathematica. Time used: 0.432 (sec). Leaf size: 54

ode=(x*Log[y[x]]+x*y[x])+(y[x]*Log[x]+x*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},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 ] \\ y(x)\to W(1) \\ \end{align*}

✓ Sympy. Time used: 0.570 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + x*log(y(x)) + (x*y(x) + y(x)*log(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \limits ^{y{\left (x \right )}} \frac {y}{y + \log {\left (y \right )}}\, dy = C_{1} - \int \frac {x}{x + \log {\left (x \right )}}\, dx \]