63.22.8 problem 4(h)

Internal problem ID [13159]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 237
Problem number : 4(h)
Date solved : Wednesday, March 05, 2025 at 09:18:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=9 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 9*y(t), diff(y(t),t) = -x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ y &= \frac {\cos \left (3 t \right ) c_{1}}{3}-\frac {\sin \left (3 t \right ) c_{2}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 42

ode={D[x[t],t]==0*x[t]+9*y[t],D[y[t],t]==-x[t]+0*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 \cos (3 t)+3 c_2 \sin (3 t) \\ y(t)\to c_2 \cos (3 t)-\frac {1}{3} c_1 \sin (3 t) \\ \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-9*y(t) + Derivative(x(t), t),0),Eq(x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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