You need to have both equations in a single DSolve function--otherwise there is no way for DSolve to know that there are two simultaneous equations. Also note that your wx'[t] should be w x'[t]: i.e., a space between the w and the x. Similarly for the wy'[t].
With that said, here is the form for DSolve that you want to use for these two simultaneous differential equations in x[t] and y[t]:
DSolve[
{y''[t] == w^2 y[t] - 2 w x'[t],
x''[t] == w^2 x[t] + 2 w y'[t]},
{y[t], x[t]}, t]
This gives a result in terms of unknown integration constants C[1], C[2], C[3], and C[4]. If you have initial conditions you can include them in the first argument list of the DSolve and the integration constants will be determined in the soluttion.
