14.32.3 problem 3

Internal problem ID [2827]
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 : 3
Date solved : Sunday, March 30, 2025 at 12:33:22 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+5 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = 4*x__1(t)-x__2(t), diff(x__2(t),t) = -2*x__1(t)+5*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{6 t}+c_2 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= -2 c_1 \,{\mathrm e}^{6 t}+c_2 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 72

ode={D[x1[t],t]==4*x1[t]-x2[t],D[x2[t],t]==-2*x1[t]+5*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{3 t} \left (c_1 \left (e^{3 t}+2\right )-c_2 \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{3 t} \left (c_2 \left (2 e^{3 t}+1\right )-2 c_1 \left (e^{3 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-4*x__1(t) + x__2(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) - 5*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{3 t} - \frac {C_{2} e^{6 t}}{2}, \ x^{2}{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{6 t}\right ] \]