7.25.10 problem 10

Internal problem ID [630]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 10
Date solved : Saturday, March 29, 2025 at 05:01:06 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=9 x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = -3*x__1(t)-2*x__2(t), diff(x__2(t),t) = 9*x__1(t)+3*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 53

ode={D[x1[t],t]==-3*x1[t]-2*x2[t],D[x2[t],t]==9*x1[t]+3*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (3 t)-\frac {1}{3} (3 c_1+2 c_2) \sin (3 t) \\ \text {x2}(t)\to c_2 \cos (3 t)+(3 c_1+c_2) \sin (3 t) \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-9*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)
 

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