44.6.9 problem 1(i)

Internal problem ID [9187]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.8. Integrating Factors. Page 32
Problem number : 1(i)
Date solved : Tuesday, September 30, 2025 at 06:12:19 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.093 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 1.175 (sec). Leaf size: 22

ode=(y[x]*Log[y[x]]-2*x*y[x])+(x+y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*y(x) + (x + y(x))*Derivative(y(x), x) + y(x)*log(y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x - log(y(x)))*y(x)/(x + y(x)) cannot be solved by the factorable group method