# Re(eigenvalue) inequality problem

Hello,
if a diff.eqn. has the characteristic equation $\lambda^2 + (3-K) \lambda + 1 = 0$
the eigenvalues solves to $\lambda=-3/2 + K/2 \pm 1/2 \sqrt{5-6K+K^2}$. No problem there. But when is the diff.eqn. asymp. stable, meaning $\Re(\lambda)<0$ ?

I can only get this far
$\Re(-3/2 + K/2 \pm 1/2*\sqrt{5-6K+K^2})<0$
$-3/2+1/2 \Re(K \pm \sqrt{5-6K+K^2})<0$

How can i find the values for K, where this inequality is true?

Thanks

CarlB
Homework Helper
My inclination would be to first consider the case when $$5-6K+K^2$$ is positive, and then consider the case when it is negative. If you separate them out, it shouldn't be too hard.

Carl

HallsofIvy
For K<= 1 or K>= 5, we need to look at all of $-3/2 + K/2 \pm 1/2 \sqrt{5-6K+K^2}$