14.20.9 problem 9

Internal problem ID [3899]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 06:58:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+13 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = 3*x__1(t)+13*x__2(t), diff(x__2(t),t) = -x__1(t)-3*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right ) \\ x_{2} \left (t \right ) &= \frac {2 c_1 \cos \left (2 t \right )}{13}-\frac {2 c_2 \sin \left (2 t \right )}{13}-\frac {3 c_1 \sin \left (2 t \right )}{13}-\frac {3 c_2 \cos \left (2 t \right )}{13} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 51

ode={D[x1[t],t]==3*x1[t]+13*x2[t],D[x2[t],t]==-1*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 (2 t)+(3 c_1+13 c_2) \sin (t) \cos (t)\\ \text {x2}(t)&\to c_2 \cos (2 t)-(c_1+3 c_2) \sin (t) \cos (t) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) - 13*x__2(t) + Derivative(x__1(t), t),0),Eq(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)
 

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