I am doing my homework and I know this is a very simple construction but I am unsure how to do this.
We are given $x''+x=0$ and are supposed to write it in the form $x'=Ax$. I see to go to $x''=-x$ but then how do I differentiate it properly? I know ultimately it ends up as a system of sines and cosines but my knowledge of solving linear systems is not that advanced and I am having trouble finding reading material on the internet. If you fully integrate the function in respect to x and t I believe it is -lnx=t. But this seems incorrect and does not give me an idea of how to construct a matrix to solve for $PDP^{-1}$. Thanks!