7.22.2 problem 12

Internal problem ID [577]
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 : 12
Date solved : Tuesday, March 04, 2025 at 11:27:05 AM
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: 0.020 (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 ) &= {\mathrm e}^{-t} c_1 +c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -{\mathrm e}^{-t} c_1 +c_2 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 68

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

✓ Sympy. Time used: 0.074 (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)
 

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