20.20.4 problem 4

Internal problem ID [3894]
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 : 4
Date solved : Tuesday, March 04, 2025 at 05:18:58 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=9 x_{1} \left (t \right )-2 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: 0.029 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = 9*x__1(t)-2*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 ) &= c_{1} {\mathrm e}^{8 t}+c_{2} {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{8 t}}{2}+5 c_{2} {\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 73

ode={D[x1[t],t]==9*x1[t]-2*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 \frac {1}{9} e^{-t} \left (c_1 \left (10 e^{9 t}-1\right )-2 c_2 \left (e^{9 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{9} e^{-t} \left (5 c_1 \left (e^{9 t}-1\right )-c_2 \left (e^{9 t}-10\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.087 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-9*x__1(t) + 2*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)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- t}}{5} + 2 C_{2} e^{8 t}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{8 t}\right ] \]