What, exactly, is the question? You have the solution. Do you want to solve it by that specific method, ?

Here your "Y" is really the pair (y, u) so you will need to apply that to both terms. You are told that and y'(0)= u(0)= 0 so Y (0)= (1, 0) and .

etc. You should be able to see pretty quickly that the first, y, component is the first n terms of the Taylor's series for cos(kt).