20.17.9 problem 9

Internal problem ID [3863]
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 : 9
Date solved : Tuesday, March 04, 2025 at 05:18:15 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.070 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 65

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

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

✓ Sympy. Time used: 0.122 (sec). Leaf size: 68

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

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