69.26.5 problem 772

Internal problem ID [18520]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 19. Basic concepts and definitions. Exercises page 199
Problem number : 772
Date solved : Sunday, October 12, 2025 at 05:33:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {{\mathrm e}^{-x \left (t \right )}}{t}\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right ) {\mathrm e}^{-y \left (t \right )}}{t} \end{align*}

✓ Maple. Time used: 0.137 (sec). Leaf size: 25

ode:=[diff(x(t),t) = exp(-x(t))/t, diff(y(t),t) = x(t)/t*exp(-y(t))]; 
dsolve(ode);
 

\begin{align*} \{x \left (t \right ) = \ln \left (\ln \left (t \right )+c_2 \right )\} \\ \left \{y \left (t \right ) &= \ln \left (\int \frac {x \left (t \right )}{t}d t +c_1 \right )\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 41

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

\begin{align*} x(t)&\to \log (t+c_1)\\ y(t)&\to \log \left (\operatorname {PolyLog}\left (2,\frac {t}{c_1}+1\right )+\log \left (-\frac {t}{c_1}\right ) \log (t+c_1)+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.772 (sec). Leaf size: 15

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

\[ \left \{x{\left (t \right )} = \log {\left (C_{1} + \log {\left (t \right )} \right )}, y{\left (t \right )} = C_{2} - \log {\left (t \right )}\right \} \]