26.1.9 problem First order with homogeneous Coefficients. Exercise 7.10, page 61

Internal problem ID [6914]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 7
Problem number : First order with homogeneous Coefficients. Exercise 7.10, page 61
Date solved : Tuesday, September 30, 2025 at 04:05:13 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.16 (sec). Leaf size: 29

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

\[ \text {Solve}\left [-2 e^{\frac {x}{y(x)}}-\log \left (\frac {y(x)}{x}\right )=\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.520 (sec). Leaf size: 14

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

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