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## Homework Statement

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.

## Homework Equations

y'=Ay

y(0)=y

_{0}

## The Attempt at a Solution

I found matrix A

[0 1

-1 0].

The eigenvalues are i and -i, and the eigenvectors

[1 -i]^T

[1 i]^T

I found the geneal solution to be:

y(t) = c

_{1}e

^{it}[1 i]^T+c

_{2}e

^{-it}[1 -i]^T

Which is equivalent,

y(t)=c

_{1}[cos(t) -sin(t)]^T + c

_{2}[sin(t) cos(t)]^T

I just don't know how to incorporate the initial conditions that y(0)=2 and y'(0)=0?

Any ideas?