76.29.1 problem 1

Internal problem ID [17798]
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 : 1
Date solved : Thursday, March 13, 2025 at 10:54:21 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 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 )-2 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 30

ode:=[diff(x__1(t),t) = 4*x__1(t)-9*x__2(t), diff(x__2(t),t) = x__1(t)-2*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (3 c_{2} t +3 c_{1} -c_{2} \right )}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 41

ode={D[x1[t],t]==4*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 e^t (3 c_1 t-9 c_2 t+c_1) \\ \text {x2}(t)\to e^t ((c_1-3 c_2) t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-4*x__1(t) + 9*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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