75.26.5 problem 772

Internal problem ID [17233]
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 : Tuesday, January 28, 2025 at 08:27:34 PM
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*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 25

dsolve([diff(x(t),t)=exp(-x(t))/t,diff(y(t),t)=x(t)/t*exp(-y(t))],singsol=all)
 

\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*}

✓ Solution by Mathematica

Time used: 0.106 (sec). Leaf size: 41

DSolve[{D[x[t],t]==Exp[-x[t]],D[y[t],t]==x[t]/t*Exp[-y[t]]},{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*}