69.8.11 problem 209

Internal problem ID [18114]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 209
Date solved : Thursday, October 02, 2025 at 02:43:15 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&={\mathrm e}^{\frac {y^{\prime }}{y}} \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 25

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

\[ y = -\operatorname {LambertW}\left (c_1 \,{\mathrm e}^{-x}\right ) {\mathrm e}^{-\frac {1}{\operatorname {LambertW}\left (c_1 \,{\mathrm e}^{-x}\right )}} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] W\left (-\frac {1}{K[1]}\right )}dK[1]\&\right ][-x+c_1] \end{align*}

✗ Sympy

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

NotImplementedError : multiple generators [_X0, exp(_X0/y(x))] 
No algorithms are implemented to solve equation _X0 - exp(_X0/y(x))