67.27.10 problem 38.10 (d)

Internal problem ID [17053]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.10 (d)
Date solved : Thursday, October 02, 2025 at 01:42:22 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.119 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 42

ode={D[x[t],t]==-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 (4 t)-\frac {1}{2} c_2 \sin (4 t)\\ y(t)&\to c_2 \cos (4 t)+2 c_1 \sin (4 t) \end{align*}

✓ Sympy. Time used: 0.036 (sec). Leaf size: 36

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

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