I need to solve this equation for $x(t)$ and $y(t)$ but I couldn't solve it without Laplace transformation.
$$\begin{cases} x'' = Ay' \\ y''=-Ax' \end{cases}$$
where A is a constant.
And how can I propose an initial condition to the solution so the solution can be a circle in $xy$-plane?
Hint Differentiating and substituting yields the equation
$$x''' = A y'' = A(-Ax') = -A^2 x' $$
in $x$. If we use the variable $x' = u$ and rearrange, this becomes the familiar, second-order linear o.d.e. $$u'' + A^2 u = 0 .$$