76.5.10 problem 10

Internal problem ID [17359]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.7 (Substitution Methods). Problems at page 108
Problem number : 10
Date solved : Monday, March 31, 2025 at 03:59:36 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.221 (sec). Leaf size: 29

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

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

✓ Sympy. Time used: 0.696 (sec). Leaf size: 12

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

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