# Thread: Proving Stability and Asymptotic Stability of Homogeneous Equations

1. ## Proving Stability and Asymptotic Stability of Homogeneous Equations

Hi,
Using these definitions:

A homogeneous equation with constant coefficients is said to be stable if all solutions remain bounded as t-->infinity

A homogeneous equation with constant coefficients is said to be asymptotically stable if all solutions converge to the zero solution as t-->infinity

I'm trying to prove these two statements:

1) The system is stable if and only if for every root $\displaystyle \lambda = \alpha + i\beta$ of the characteristic polynomial, $\displaystyle 0 \geq \alpha$ and $\displaystyle \alpha < 0$ if the multiplicity of lambda is bigger than one.

2) The system is aymptotically stable if and only if for every root $\displaystyle \lambda = \alpha + i\beta$ of the characteristic polynomial, $\displaystyle \alpha < 0$.

Thanks a lot!

2. ...When is, for fixed n, the limit $\displaystyle \lim_{x\rightarrow+\infty}x^n\exp(\alpha x)\cos(\beta x)$ zero? And when is it infinite?