# First order system ODE, complex root

1. Dec 3, 2012

### kougou

Hi.

For first order system ODE, complex root.
y'=Ay, where A is a 2by2 matrix. I am assuming the roots are complex. After finding the eigenvalue (complex conjugate) and their eigen-vectors (which come in a form of complex conjugate again), we plug into the solution y=ζexp(λt), where λ is egenvalue and ζ is its corresponding eigenvector. And Since λ is complex, we apply euler's formula.And at the end, we get to the part
y= u(t) + i*v(t).

I don't understand why both u(t) and v(t) are solution to the system. u(t) is clear. But v(t) is not. what happen to its imaginary number?

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Last edited: Dec 3, 2012