Internal problem ID [15499]
Internal file name [OUTPUT/15500_Friday_May_10_2024_05_47_29_PM_99659885/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: 768.
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_{1}^{\prime }\left (t \right )&={\mathrm e}^{t} {\mathrm e}^{-x_{1} \left (t \right )}\\ x_{2}^{\prime }\left (t \right )&=2 \,{\mathrm e}^{x_{1} \left (t \right )} \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 24
dsolve([diff(x__1(t),t)=exp(t-x__1(t)),diff(x__2(t),t)=2*exp(x__1(t))],singsol=all)
\begin{align*} \{x_{1} \left (t \right ) &= \ln \left ({\mathrm e}^{t}+c_{2} \right )\} \\ \{x_{2} \left (t \right ) &= \int 2 \,{\mathrm e}^{x_{1} \left (t \right )}d t +c_{1}\} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.15 (sec). Leaf size: 28
DSolve[{x1'[t]==Exp[t-x1[t]],x2'[t]==2*Exp[x1[t]]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \log \left (e^t+c_1\right ) \\ \text {x2}(t)\to 2 e^t+2 c_1 t+c_2 \\ \end{align*}