78.4.14 problem 15

Internal problem ID [18066]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 15
Date solved : Monday, March 31, 2025 at 05:02:44 PM
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: 0.004 (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);
 

\[ \int \frac {x}{\ln \left (x \right )+x}d x +\int _{}^{y}\frac {\textit {\_a}}{\ln \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.58 (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: 0.605 (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)
 

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