20.16.16 problem 16

Internal problem ID [3849]
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 : 16
Date solved : Tuesday, March 04, 2025 at 05:18:01 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 51

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = -x__1(t), diff(x__3(t),t) = -x__4(t), diff(x__4(t),t) = x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \sin \left (t \right )+c_4 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= \cos \left (t \right ) c_3 -c_4 \sin \left (t \right ) \\ x_{3} \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ x_{4} \left (t \right ) &= \sin \left (t \right ) c_{2} -\cos \left (t \right ) c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 60

ode={D[x1[t],t]==x2[t],D[x2[t],t]==-x1[t],D[x3[t],t]==-x4[t],D[x4[t],t]==x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (t)+c_2 \sin (t) \\ \text {x2}(t)\to c_2 \cos (t)-c_1 \sin (t) \\ \text {x3}(t)\to c_3 \cos (t)-c_4 \sin (t) \\ \text {x4}(t)\to c_4 \cos (t)+c_3 \sin (t) \\ \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(-x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) + Derivative(x__2(t), t),0),Eq(x__4(t) + Derivative(x__3(t), t),0),Eq(-x__3(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )}, \ x^{3}{\left (t \right )} = - C_{3} \sin {\left (t \right )} - C_{4} \cos {\left (t \right )}, \ x^{4}{\left (t \right )} = C_{3} \cos {\left (t \right )} - C_{4} \sin {\left (t \right )}\right ] \]