20.19.1 problem 2

Internal problem ID [3881]
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 : 2
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 )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 29

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

\begin{align*} \text {x1}(t)\to e^{2 t} (c_2 t+c_1) \\ \text {x2}(t)\to c_2 e^{2 t} \\ \end{align*}

✓ Sympy. Time used: 0.082 (sec). Leaf size: 26

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

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