20.16.8 problem 8

Internal problem ID [3841]
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 : 8
Date solved : Tuesday, March 04, 2025 at 05:17:52 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 )&=5 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 35

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = -x__1(t), diff(x__3(t),t) = 5*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ x_{2} \left (t \right ) &= \cos \left (t \right ) c_{1} -\sin \left (t \right ) c_{2} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 76

ode={D[x1[t],t]==x2[t],D[x2[t],t]==-x1[t],D[x3[t],t]==5*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[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 e^{5 t} \\ \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 0 \\ \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) + Derivative(x__2(t), t),0),Eq(-5*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )}, \ x^{3}{\left (t \right )} = C_{3} e^{5 t}\right ] \]