69.26.2 problem 768

Internal problem ID [18517]
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 : Sunday, October 12, 2025 at 05:33:58 AM
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*}

✓ Maple. Time used: 0.140 (sec). Leaf size: 23

ode:=[diff(x__1(t),t) = exp(t-x__1(t)), diff(x__2(t),t) = 2*exp(x__1(t))]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 28

ode={D[ x1[t],t]==Exp[t-x1[t]],D[ x2[t],t]==2*Exp[x1[t]]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-exp(t - x__1(t)) + Derivative(x__1(t), t),0),Eq(-2*exp(x__1(t)) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

NotImplementedError :