75.26.2 problem 768

Internal problem ID [17230]
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
Date solved : Tuesday, January 28, 2025 at 08:27:33 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&={\mathrm e}^{t -x_{1} \left (t \right )}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 \,{\mathrm e}^{x_{1} \left (t \right )} \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 23

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: 0.122 (sec). Leaf size: 28

DSolve[{D[ x1[t],t]==Exp[t-x1[t]],D[ x2[t],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*}