78.4.14 problem 15

Internal problem ID [18137]
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 : Tuesday, January 28, 2025 at 11:31:42 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.004 (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)
 

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

✓ Solution by Mathematica

Time used: 0.580 (sec). Leaf size: 54

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