7.22.5 problem 15

Internal problem ID [580]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.1 (First order systems and applications). Problems at page 335
Problem number : 15
Date solved : Saturday, March 29, 2025 at 04:57:11 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {y \left (t \right )}{2}\\ \frac {d}{d t}y \left (t \right )&=-8 x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 1/2*y(t), diff(y(t),t) = -8*x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right ) \\ y \left (t \right ) &= 4 c_1 \cos \left (2 t \right )-4 c_2 \sin \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 42

ode={D[x[t],t]==1/2*y[t],D[y[t],t]==-8*x[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 \cos (2 t)+\frac {1}{4} c_2 \sin (2 t) \\ y(t)\to c_2 \cos (2 t)-4 c_1 \sin (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.073 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t)/2 + Derivative(x(t), t),0),Eq(8*x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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