76.29.2 problem 2

Internal problem ID [17807]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.7 (Defective Matrices). Problems at page 444
Problem number : 2
Date solved : Monday, March 31, 2025 at 04:33:02 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-9 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.088 (sec). Leaf size: 23

ode:=[diff(x__1(t),t) = 3*x__1(t)-9*x__2(t), diff(x__2(t),t) = x__1(t)-3*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 t +c_2 \\ x_{2} \left (t \right ) &= -\frac {1}{9} c_1 +\frac {1}{3} c_1 t +\frac {1}{3} c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 114

ode={D[x1[t],t]==3*x1[t]-9*x2[t],D[x2[t],t]==1*x1[t]-2*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{11} e^{t/2} \left (11 c_1 \cos \left (\frac {\sqrt {11} t}{2}\right )+\sqrt {11} (5 c_1-18 c_2) \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ \text {x2}(t)\to \frac {1}{11} e^{t/2} \left (11 c_2 \cos \left (\frac {\sqrt {11} t}{2}\right )+\sqrt {11} (2 c_1-5 c_2) \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.065 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + 9*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 3*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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