72.11.1 problem 3

Internal problem ID [14755]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.4 page 310
Problem number : 3
Date solved : Thursday, March 13, 2025 at 04:18:19 AM
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: 0.038 (sec). Leaf size: 17

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 &= -\sin \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 18

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: 0.062 (sec). Leaf size: 31

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)
 

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