Internal problem ID [15502]
Internal file name [OUTPUT/15503_Friday_May_10_2024_05_47_30_PM_79597876/index.tex]
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.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs" Unable to solve or complete the solution.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {{\mathrm e}^{-x \left (t \right )}}{t}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right ) {\mathrm e}^{-y \left (t \right )}}{t} \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 26
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: 0.107 (sec). Leaf size: 41
DSolve[{x'[t]==Exp[-x[t]],y'[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*}