74.22.6 problem 6

Internal problem ID [16582]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 6. Systems of Differential Equations. Exercises 6.1, page 282
Problem number : 6
Date solved : Monday, March 31, 2025 at 02:59:36 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 17

ode:=[diff(x__1(t),t) = -x__1(t)+1, diff(x__2(t),t) = x__2(t)]; 
ic:=x__1(0) = 0x__2(0) = 1; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 12

ode={D[ x1[t],t]==-x1[t],D[ x2[t],t]==x2[t]}; 
ic={x1[0]==0,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 0 \\ \text {x2}(t)\to e^t \\ \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t) + Derivative(x__1(t), t) - 1,0),Eq(-x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- t} + 1, \ x^{2}{\left (t \right )} = C_{2} e^{t}\right ] \]