66.3.1 problem Problem 1

Internal problem ID [13944]
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 : Tuesday, January 28, 2025 at 06:09:19 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*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

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

dsolve([diff(x(t),t) = y(t), diff(y(t),t) = -x(t), x(0) = 0, y(0) = 1], singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==y[t],D[y[t],t]==-x[t]},{},{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*}