20.17.5 problem 5

Internal problem ID [3859]
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 : 5
Date solved : Tuesday, March 04, 2025 at 05:18:12 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 46

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

\begin{align*} x_{1} \left (t \right ) &= c_{1} +c_{2} t +{\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= c_{2} +\frac {2 \,{\mathrm e}^{2 t} c_3}{3} \\ x_{3} \left (t \right ) &= c_{2} +\frac {4 \,{\mathrm e}^{2 t} c_3}{3}+2 c_{1} +2 c_{2} t \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 138

ode={D[x1[t],t]==2*x1[t]+2*x2[t]-x3[t],D[x2[t],t]==2*x1[t]+x2[t]-x3[t],D[x3[t],t]==2*x1[t]+3*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} \left (2 c_1 \left (-2 t+3 e^{2 t}-1\right )+c_2 \left (2 t+3 e^{2 t}-3\right )+c_3 \left (2 t-3 e^{2 t}+3\right )\right ) \\ \text {x2}(t)\to \frac {1}{2} \left (2 c_1 \left (e^{2 t}-1\right )+c_2 e^{2 t}-c_3 e^{2 t}+c_2+c_3\right ) \\ \text {x3}(t)\to 2 c_1 \left (-t+e^{2 t}-1\right )+c_2 \left (t+e^{2 t}-1\right )+c_3 \left (t-e^{2 t}+2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 49

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) - 2*x__2(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-2*x__1(t) - 3*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)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} t - \frac {C_{1}}{2} + C_{2} + \frac {3 C_{3} e^{2 t}}{4}, \ x^{2}{\left (t \right )} = C_{1} + \frac {C_{3} e^{2 t}}{2}, \ x^{3}{\left (t \right )} = 2 C_{1} t + 2 C_{2} + C_{3} e^{2 t}\right ] \]