20.17.1 problem 1

Internal problem ID [3855]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.5 (Defective coefficient matrix), page 619
Problem number : 1
Date solved : Tuesday, March 04, 2025 at 05:18:08 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 28

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t), diff(x__2(t),t) = -x__1(t)+3*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 ) &= {\mathrm e}^{2 t} \left (c_{2} t +c_{1} +c_{2} \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 44

ode={D[x1[t],t]==x1[t]+x2[t],D[x2[t],t]==-x1[t]+3*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.099 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 3*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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