7.22.3 problem 13

Internal problem ID [578]
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 : 13
Date solved : Saturday, March 29, 2025 at 04:57:09 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 )&=2 x \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: 0.149 (sec). Leaf size: 15

ode:=[diff(x(t),t) = -2*y(t), diff(y(t),t) = 2*x(t)]; 
ic:=x(0) = 1y(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= \cos \left (2 t \right ) \\ y \left (t \right ) &= \sin \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 16

ode={D[x[t],t]==-2*y[t],D[y[t],t]==2*x[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \cos (2 t) \\ y(t)\to \sin (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 32

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

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