Dear all,

I have a question on the stability of a solution.

Let a(t), b(t) and c(t) be continuous functions of t over the interval $\displaystyle [0,\infty)$. Assume (x,y) = $\displaystyle (\phi(t), \psi(t))$ is a solution of the system

$\displaystyle \dot{x} = -a^2(t)y + b(t), \dot{y} = a^2(t)x + c(t)$

Show that this solution is stable.

I rearranged the system to get

$\displaystyle \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 $\displaystyle 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 $\displaystyle a^4(t)$ and trace(A) = 0 by treating t as constant. Also, what role does the $\displaystyle \binom{b(t)}{c(t)}$ play here?

Can someone please help me?

Thank you.

Regards,

Rayne