60.1.207 problem 211

Internal problem ID [10221]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 211
Date solved : Sunday, March 30, 2025 at 03:32:17 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} y y^{\prime }-x \,{\mathrm e}^{\frac {x}{y}}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 31

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

\[ y = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}}{-\textit {\_a}^{2}+{\mathrm e}^{\frac {1}{\textit {\_a}}}}d \textit {\_a} +\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 41

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]}{K[1]^2-e^{\frac {1}{K[1]}}}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.516 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {u_{1} e^{u_{1}}}{u_{1}^{2} e^{u_{1}} - 1}\, du_{1}} \]