14.32.10 problem 10

Internal problem ID [2834]
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 : 10
Date solved : Sunday, March 30, 2025 at 12:33:31 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-9 x_{1} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = 4*x__2(t), diff(x__2(t),t) = -9*x__1(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \sin \left (6 t \right )+c_2 \cos \left (6 t \right ) \\ x_{2} \left (t \right ) &= \frac {3 c_1 \cos \left (6 t \right )}{2}-\frac {3 c_2 \sin \left (6 t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 44

ode={D[x1[t],t]==0*x1[t]+4*x2[t],D[x2[t],t]==-9*x1[t]-0*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (6 t)+\frac {2}{3} c_2 \sin (6 t) \\ \text {x2}(t)\to c_2 \cos (6 t)-\frac {3}{2} c_1 \sin (6 t) \\ \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-4*x__2(t) + Derivative(x__1(t), t),0),Eq(9*x__1(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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