# Stability of a solution

1. Sep 6, 2007

### wu_weidong

1. The problem statement, all variables and given/known data
Let a(t), b(t) and c(t) be continuous functions of t over the interval $$[0,\infty)$$. Assume (x,y) = $$(\phi(t), \psi(t))$$ is a solution of the system
$$\dot{x} = -a^2(t)y + b(t), \dot{y} = a^2(t)x + c(t)$$

Show that this solution is stable.

3. The attempt at a solution
I rearranged the system to get

$$\frac{d}{dt}\binom{x}{y} = \binom{-a^2(t)y + b(t)}{a^2(t)x + c(t)} = \left( \begin{array}{cc} 0 & -a^2(t) \\ a^2(t) & 0 \end{array} \right)\binom{x}{y} + \binom{b(t)}{c(t)}$$

I've only dealt with constant coefficient linear systems before, so I'm having trouble with this question.

Let $$A = \left( \begin{array}{cc} 0 & -a^2(t) \\ a^2(t) & 0 \end{array} \right)$$.

I'm not sure if I can treat A like a constant matrix, i.e. take the determinant of A to be $$a^4(t)$$ and trace(A) = 0 by treating t as constant. Also, what role does the $$\binom{b(t)}{c(t)}$$ play here?

Thank you.

Regards,
Rayne

Last edited: Sep 6, 2007
2. Sep 6, 2007

3. Sep 7, 2007

### wu_weidong

Do I need to find a Lyapunov function?

4. Sep 7, 2007

### Mr.Brown

Couldn´t you just find an integrating factor h integrate the whole thing and find an integral expression for the whole thing.
Then you got the time T operator differentiate ( take integral sign away) and show that your solution is an eigenvector with Eigenvalue(floquet multiplier) = 1 for the system?