85.83.1 problem 1 (a)

Internal problem ID [22998]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 444
Problem number : 1 (a)
Date solved : Thursday, October 02, 2025 at 09:17:24 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 11

ode:=[diff(y(t),t) = x(t), diff(x(t),t) = -y(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \cos \left (t \right ) \\ y \left (t \right ) &= \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 12

ode={D[y[t],t]==x[t],D[x[t],t]==-y[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \cos (t)\\ y(t)&\to \sin (t) \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 17

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

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