14.32.2 problem 2

Internal problem ID [2826]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.7 (Phase portraits of linear systems). Page 427
Problem number : 2
Date solved : Tuesday, March 04, 2025 at 02:50:11 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )-6 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = -x__2(t), diff(x__2(t),t) = 8*x__1(t)-6*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{-4 t} \\ x_{2} \left (t \right ) &= 2 c_1 \,{\mathrm e}^{-2 t}+4 c_2 \,{\mathrm e}^{-4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 70

ode={D[x1[t],t]==0*x1[t]-x2[t],D[x2[t],t]==8*x1[t]-6*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-4 t} \left (c_1 \left (4 e^{2 t}-2\right )-c_2 \left (e^{2 t}-1\right )\right ) \\ \text {x2}(t)\to e^{-4 t} \left (4 c_1 \left (e^{2 t}-1\right )-c_2 \left (e^{2 t}-2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__2(t) + Derivative(x__1(t), t),0),Eq(-8*x__1(t) + 6*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- 4 t}}{4} + \frac {C_{2} e^{- 2 t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{- 2 t}\right ] \]