Determine the General solution to the differential Equation system X'=AX
For a simple problem like this, danny arrigo's suggestion is best, but you can do as a matrix problem.
Find the eigenvalues of . It should be easy to see that they are 4i and -4i. Since those are distinct, the matrix can be diagonalize (over the complex numbers).
With eigenvalue 4i, we have which gives the two equations -4y= 4ix and 4x= 4iy. Those both reduce to y=ix so one eigenvector is <1, i>.
With eigenvalue -4i, we have which gives the two equations -4y= -4ix and 4x= -4iy. Those both reduce to y=-ix so another, independent, eigenvector is <1, i>.
If we use those as columns in matrix P, then and
Then it is easy to see that .
If we multiply, on the left, by the constant matrix , we have .
Let so that and substitue PY for X in the equation: so our differential equation becomes .
The general solution to that, just like any y'= ay, is and because that is a diagonal matrix it is easy to show that we get
Now we use to write .
And, then use the fact that , to change into sine and cosine, getting danny aririgo's answer.
Whew! Well, I said that, for a simple problem like this danny arrigo's solution was best!