20.16.20 problem 20

Internal problem ID [3853]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 20
Date solved : Tuesday, March 04, 2025 at 05:18:05 PM
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 )&=-4 x_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 27

ode:=[diff(x__1(t),t) = 4*x__2(t), diff(x__2(t),t) = -4*x__1(t)]; 
ic:=x__1(0) = 1x__2(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \sin \left (4 t \right )+\cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= \cos \left (4 t \right )-\sin \left (4 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 28

ode={D[x1[t],t]==4*x2[t],D[x2[t],t]==-4*x1[t]}; 
ic={x1[0]==1,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \sin (4 t)+\cos (4 t) \\ \text {x2}(t)\to \cos (4 t)-\sin (4 t) \\ \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 31

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(4*x__1(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} \sin {\left (4 t \right )} + C_{2} \cos {\left (4 t \right )}, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (4 t \right )} - C_{2} \sin {\left (4 t \right )}\right ] \]