78.5.10 problem 2 (j)

Internal problem ID [18153]
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 (j)
Date solved : Tuesday, January 28, 2025 at 11:34:18 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{2}+y x +1+\left (x^{2}+y x +1\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 27

dsolve(( y(x)^2+x*y(x)+1 ) +( x^2+x*y(x)+1 )*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = \frac {-x^{2}+\operatorname {LambertW}\left (-2 x c_1 \,{\mathrm e}^{\left (x -1\right ) \left (x +1\right )}\right )}{x} \]

✓ Solution by Mathematica

Time used: 5.421 (sec). Leaf size: 56

DSolve[( y[x]^2+x*y[x]+1 )+( x^2+x*y[x]+1 )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x+\frac {W\left (x \left (-e^{x^2-1+c_1}\right )\right )}{x} \\ y(x)\to -x \\ y(x)\to \frac {W\left (-e^{x^2-1} x\right )}{x}-x \\ \end{align*}