20.17.3 problem 3

Internal problem ID [3857]
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 : 3
Date solved : Tuesday, March 04, 2025 at 05:18:10 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 32

ode:=[diff(x__1(t),t) = -3*x__1(t)-2*x__2(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)]; 
dsolve(ode);
 

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

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

ode={D[x1[t],t]==-3*x1[t]-2*x2[t],D[x2[t],t]==2*x1[t]+x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.082 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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