78.8.9 problem 9

Internal problem ID [18207]
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. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 11:38:22 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x y y^{\prime }&=x^{2} y^{\prime }+y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

Time used: 1.902 (sec). Leaf size: 25

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

\begin{align*} y(x)\to -x W\left (-\frac {e^{-c_1}}{x}\right ) \\ y(x)\to 0 \\ \end{align*}