14.32.6 problem 6

Internal problem ID [2830]
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 : 6
Date solved : Sunday, March 30, 2025 at 12:33:26 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = 3*x__1(t)-x__2(t), diff(x__2(t),t) = 5*x__1(t)-3*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+5 c_2 \,{\mathrm e}^{-2 t} \\ \end{align*}

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

ode={D[x1[t],t]==3*x1[t]-x2[t],D[x2[t],t]==5*x1[t]-3*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-2 t} \left (c_1 \left (5 e^{4 t}-1\right )-c_2 \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-2 t} \left (5 c_1 \left (e^{4 t}-1\right )-c_2 \left (e^{4 t}-5\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + x__2(t) + Derivative(x__1(t), t),0),Eq(-5*x__1(t) + 3*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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