78.5.9 problem 2 (i)

Internal problem ID [18152]
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 9 (Integrating Factors). Problems at page 80
Problem number : 2 (i)
Date solved : Tuesday, January 28, 2025 at 11:34:16 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y \ln \left (y\right )-2 y x +\left (x +y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 36

dsolve(( y(x)*ln(y(x)) -2*x*y(x) ) +( x+y(x)  )*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{\frac {x^{2}-x \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {x^{2}-c_1}{x}}}{x}\right )-c_1}{x}} \]

✓ Solution by Mathematica

Time used: 1.095 (sec). Leaf size: 22

DSolve[( y[x]*Log[y[x]] -2*x*y[x] )+( x+y[x] )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x W\left (\frac {e^{x+\frac {c_1}{x}}}{x}\right ) \]