87.29.10 problem 10

Internal problem ID [23890]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 304
Problem number : 10
Date solved : Thursday, October 02, 2025 at 09:46:19 PM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 20

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

\begin{align*} x(t)&\to \sin (t)+\cos (t)\\ y(t)&\to \sin (t)-\cos (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(y(t) + Derivative(x(t), t),0),Eq(-x(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): -1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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