87.30.2 problem 2

Internal problem ID [23901]
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 310
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:46:23 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*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 30

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{-t}-c_2 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 70

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

\begin{align*} x(t)&\to \frac {1}{2} e^{-t} \left (c_1 e^{2 t}-c_2 e^{2 t}+c_1+c_2\right )\\ y(t)&\to \frac {1}{2} e^{-t} \left (c_1 \left (-e^{2 t}\right )+c_2 e^{2 t}+c_1+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.040 (sec). Leaf size: 24

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 = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- t} - C_{2} e^{t}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t}\right ] \]