solution to a differential equation I have given the following differential equation:
$x'= - y$ and $y' = x$
How can I solve them?
Thanks for helping!
Greetings

Let $\displaystyle X(t)= \binom{x(t)}{y(t)}$ so
$$ X' = \left( \begin{array}{ccc}
0 & -1  \\
1 & 0  \\
\end{array} \right)X .$$
This has solution $$ X(t)=  \exp\biggr(  \left( \begin{array}{ccc}
0 & -t  \\
t & 0  \\
\end{array} \right) \biggr) X(0)=   \left( \begin{array}{ccc}
0 & e^{-t}  \\
e^t & 0  \\
\end{array} \right)\binom{x(0)}{y(0)}$$
so $$ x(t) = y(0) e^{-t}  \ \ \text{ and } \ \   y(t) = x(0) e^{t} . $$
