75.12.18 problem 292

Internal problem ID [16813]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 292
Date solved : Monday, March 31, 2025 at 03:22:33 PM
CAS classification : [[_homogeneous, `class A`], _exact, _dAlembert]

\begin{align*} 1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=1 \end{align*}

✓ Maple. Time used: 0.229 (sec). Leaf size: 21

ode:=1+exp(x/y(x))+exp(x/y(x))*(1-x/y(x))*diff(y(x),x) = 0; 
ic:=y(1) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.338 (sec). Leaf size: 21

ode=(1+Exp[x/y[x]])+(Exp[x/y[x]]*(1-x/y[x]))*D[y[x],x]==0; 
ic={y[1]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {x}{W\left (\frac {x}{x-e-1}\right )} \]

✓ Sympy. Time used: 2.364 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x/y(x) + 1)*exp(x/y(x))*Derivative(y(x), x) + exp(x/y(x)) + 1,0) 
ics = {y(1): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x}{W\left (\frac {x}{x - e - 1}\right )} \]