66.3.1 problem Problem 1

Internal problem ID [13865]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 3, SYSTEMS OF DIFFERENTIAL EQUATIONS. Problems page 209
Problem number : Problem 1
Date solved : Wednesday, March 05, 2025 at 10:19:59 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 ) = 0\\ y \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 11

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 31

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 c_1 \cos (t)+c_2 \sin (t) \\ y(t)\to c_2 \cos (t)-c_1 \sin (t) \\ \end{align*}

✓ Sympy. Time used: 0.062 (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)
 

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