20.20.6 problem 6

Internal problem ID [3896]
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 : 6
Date solved : Tuesday, March 04, 2025 at 05:19:00 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-8 x_{1} \left (t \right )+5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-5 x_{1} \left (t \right )+2 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 32

ode:=[diff(x__1(t),t) = -8*x__1(t)+5*x__2(t), diff(x__2(t),t) = -5*x__1(t)+2*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{2} t +c_{1} \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (5 c_{2} t +5 c_{1} +c_{2} \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 46

ode={D[x1[t],t]==-8*x1[t]+5*x2[t],D[x2[t],t]==-5*x1[t]+2*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} (-5 c_1 t+5 c_2 t+c_1) \\ \text {x2}(t)\to e^{-3 t} (-5 c_1 t+5 c_2 t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(8*x__1(t) - 5*x__2(t) + Derivative(x__1(t), t),0),Eq(5*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)
 

\[ \left [ x^{1}{\left (t \right )} = - 5 C_{2} t e^{- 3 t} - \left (5 C_{1} - C_{2}\right ) e^{- 3 t}, \ x^{2}{\left (t \right )} = - 5 C_{1} e^{- 3 t} - 5 C_{2} t e^{- 3 t}\right ] \]