20.16.2 problem 2

Internal problem ID [3835]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 2
Date solved : Tuesday, March 04, 2025 at 05:17:46 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = -2*x__1(t)-7*x__2(t), diff(x__2(t),t) = -x__1(t)+4*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{5 t} \\ x_{2} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{-3 t}}{7}-c_{2} {\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

ode={D[x1[t],t]==-2*x1[t]-7*x2[t],D[x2[t],t]==-x1[t]+4*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{8} e^{-3 t} \left (c_1 \left (e^{8 t}+7\right )-7 c_2 \left (e^{8 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{8} e^{-3 t} \left (c_1 \left (-e^{8 t}\right )+7 c_2 e^{8 t}+c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(2*x__1(t) + 7*x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 7 C_{1} e^{- 3 t} - C_{2} e^{5 t}, \ x^{2}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{5 t}\right ] \]