44.5.5 problem 1(e)

Internal problem ID [9161]
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 1.7. Homogeneous Equations. Page 28
Problem number : 1(e)
Date solved : Tuesday, September 30, 2025 at 06:10:49 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y^{\prime }&=y+2 x \,{\mathrm e}^{-\frac {y}{x}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

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

\[ y = \left (\ln \left (2\right )+\ln \left (\ln \left (x \right )+c_1 \right )\right ) x \]

✓ Mathematica. Time used: 0.258 (sec). Leaf size: 15

ode=x*D[y[x],x]==y[x]+2*x*Exp[-y[x]/x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x \log (2 \log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.357 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \log {\left (\left (C_{1} + \log {\left (x^{2} \right )}\right )^{x} \right )} \]