The higher order equation y"+y=0 can be written as a unknown d/dt[y y']=[y' y"]=[y' -y]
If this is du/dt=Au, what is the 2x2 matrix A? Find its eigenvectors and eigenvalues, and compute the solution THAT STARTS FROM y(0)=2, y'(0)=0.
The Attempt at a Solution
I found matrix A
The eigenvalues are i and -i, and the eigenvectors
I found the geneal solution to be:
y(t) = c1eit[1 i]^T+c2e-it[1 -i]^T
Which is equivalent,
y(t)=c1[cos(t) -sin(t)]^T + c2[sin(t) cos(t)]^T
I just don't know how to incorporate the initial conditions that y(0)=2 and y'(0)=0???