20.17.4 problem 4

Internal problem ID [3858]
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 : 4
Date solved : Tuesday, March 04, 2025 at 05:18:11 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}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: 0.037 (sec). Leaf size: 81

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),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 ) &= {\mathrm e}^{t} c_{1} -c_{2} {\mathrm e}^{-t}-c_3 \,{\mathrm e}^{-t} t -c_3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} c_{1} +c_{2} {\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-t} t \\ x_{3} \left (t \right ) &= {\mathrm e}^{t} c_{1} -c_{2} {\mathrm e}^{-t}-c_3 \,{\mathrm e}^{-t} t +c_3 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 152

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

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left (c_1 \left (2 t+e^{2 t}+3\right )+2 c_2 \left (e^{2 t}-1\right )+c_3 \left (-2 t+e^{2 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-t} \left (c_1 \left (-2 t+e^{2 t}-1\right )+2 c_2 \left (e^{2 t}+1\right )+c_3 \left (2 t+e^{2 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{4} e^{-t} \left (c_1 \left (2 t+e^{2 t}-1\right )+2 c_2 \left (e^{2 t}-1\right )+c_3 \left (-2 t+e^{2 t}+3\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__2(t) + Derivative(x__1(t), t),0),Eq(-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)
 

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