20.17.6 problem 6

Internal problem ID [3860]
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 : 6
Date solved : Tuesday, March 04, 2025 at 05:18:13 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 41

ode:=[diff(x__1(t),t) = -2*x__1(t), diff(x__2(t),t) = x__1(t)-3*x__2(t)-x__3(t), diff(x__3(t),t) = -x__1(t)+x__2(t)-x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \,{\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_{2} t +c_{1} \right ) \\ x_{3} \left (t \right ) &= -{\mathrm e}^{-2 t} \left (c_{2} t -c_3 +c_{1} +c_{2} \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 61

ode={D[x1[t],t]==-2*x1[t]+0*x2[t]-0*x3[t],D[x2[t],t]==1*x1[t]-3*x2[t]-x3[t],D[x3[t],t]==-1*x1[t]+1*x2[t]-1*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{-2 t} \\ \text {x2}(t)\to e^{-2 t} ((c_1-c_2-c_3) t+c_2) \\ \text {x3}(t)\to e^{-2 t} ((-c_1+c_2+c_3) t+c_3) \\ \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(2*x__1(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 3*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(x__1(t) - x__2(t) + x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \left (C_{1} + C_{2}\right ) e^{- 2 t}, \ x^{2}{\left (t \right )} = C_{1} t e^{- 2 t} + \left (C_{2} + C_{3}\right ) e^{- 2 t}, \ x^{3}{\left (t \right )} = - C_{1} t e^{- 2 t} - C_{3} e^{- 2 t}\right ] \]