We call orbit associated to the initial datum $(t_0,x_0)$, the set of points:
$C=\{$x$(t;t_0,x_0), t \in T\}$.
Given the Matrix A:
\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
Write the equation of the orbit of the following ODE:
x'(t)$=A$x(t)
The solution of the system is :
$\begin{cases} x(t)=x_0cos(t)+y_0sin(t)\\y(t)=−x_0sin(t)+y_0cos(t)\end{cases}$
The equation of the orbit should be: $x_2^2+y_2^2=x_0^2+y_0^2$, but how do I get a that from the solution of the system?
Compute $x(t)^2$ and $y(t)^2$ and observe that $\cos^2 (t)+ \sin^2 (t)=1.$ Then you should get
$$x(t)^2+y(t)^2=x_0^2+y_0^2.$$