20.19.2 problem 3

Internal problem ID [3882]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 3
Date solved : Tuesday, March 04, 2025 at 05:18:36 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 35

ode={D[x1[t],t]==x1[t]+2*x2[t],D[x2[t],t]==-x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to (c_1+c_2) e^t-c_2 e^{-t} \\ \text {x2}(t)\to c_2 e^{-t} \\ \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 19

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

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