9.4.10 problem problem 10

Internal problem ID [974]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 10
Date solved : Tuesday, March 04, 2025 at 12:06:39 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: 0.005 (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 ) &= \sin \left (3 t \right ) c_1 +\cos \left (3 t \right ) c_2 \\ x_{2} \left (t \right ) &= -\frac {3 \cos \left (3 t \right ) c_1}{2}+\frac {3 \sin \left (3 t \right ) c_2}{2}-\frac {3 \sin \left (3 t \right ) c_1}{2}-\frac {3 \cos \left (3 t \right ) c_2}{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: 0.092 (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)
 

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