76.29.10 problem 10

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 : 10
Date solved : Thursday, March 13, 2025 at 10:54:31 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -5\\ x_{2} \left (0\right ) = 7 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 24

ode:=[diff(x__1(t),t) = 3*x__1(t)-4*x__2(t), diff(x__2(t),t) = x__1(t)-x__2(t)]; 
ic:=x__1(0) = -5x__2(0) = 7; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (-38 t -5\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-76 t +28\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 27

ode={D[x1[t],t]==3*x1[t]-4*x2[t],D[x2[t],t]==1*x1[t]-1*x2[t]}; 
ic={x1[0]==-5,x2[0]==7}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -e^t (38 t+5) \\ \text {x2}(t)\to e^t (7-19 t) \\ \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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