20.20.3 problem 3

Internal problem ID [3893]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 3
Date solved : Tuesday, March 04, 2025 at 05:18:57 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-6 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{1} \left (t \right )-5 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = -6*x__1(t)+x__2(t), diff(x__2(t),t) = 6*x__1(t)-5*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-8 t} c_{1} +c_{2} {\mathrm e}^{-3 t} \\ x_{2} \left (t \right ) &= -2 \,{\mathrm e}^{-8 t} c_{1} +3 c_{2} {\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 73

ode={D[x1[t],t]==-6*x1[t]+x2[t],D[x2[t],t]==6*x1[t]-5*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{5} e^{-8 t} \left (c_1 \left (2 e^{5 t}+3\right )+c_2 \left (e^{5 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-8 t} \left (6 c_1 \left (e^{5 t}-1\right )+c_2 \left (3 e^{5 t}+2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(6*x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(-6*x__1(t) + 5*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- 8 t}}{2} + \frac {C_{2} e^{- 3 t}}{3}, \ x^{2}{\left (t \right )} = C_{1} e^{- 8 t} + C_{2} e^{- 3 t}\right ] \]