14.32.9 problem 9

Internal problem ID [2833]
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 : 9
Date solved : Tuesday, March 04, 2025 at 02:50:19 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 36

ode:=[diff(x__1(t),t) = 2*x__1(t)+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 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= \cos \left (t \right ) c_1 -2 c_2 \cos \left (t \right )-2 c_1 \sin \left (t \right )-\sin \left (t \right ) c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 41

ode={D[x1[t],t]==2*x1[t]+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 c_1 \cos (t)+(2 c_1+c_2) \sin (t) \\ \text {x2}(t)\to c_2 (\cos (t)-2 \sin (t))-5 c_1 \sin (t) \\ \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 37

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

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