76.29.9 problem 9

Internal problem ID [17814]
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 : 9
Date solved : Monday, March 31, 2025 at 04:33:13 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-7 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 7\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 28

ode:=[diff(x__1(t),t) = x__1(t)-4*x__2(t), diff(x__2(t),t) = 4*x__1(t)-7*x__2(t)]; 
ic:=x__1(0) = 7x__2(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (24 t +7\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (96 t +4\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 30

ode={D[x1[t],t]==1*x1[t]-4*x2[t],D[x2[t],t]==4*x1[t]-7*x2[t]}; 
ic={x1[0]==7,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} (24 t+7) \\ \text {x2}(t)\to e^{-3 t} (24 t+1) \\ \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) + 7*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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