I have a problem related to a topic in electronics (filter design) which I have been studying.

Basically, I'm considering 3 unknown variables $\displaystyle a_1, a_2, b_1$ and one constant $\displaystyle c$.

From two different starting points I can workout 3 equations which should be enough to solve for $\displaystyle a_1, a_2, b_1$. However, I'm having difficulties trying to establish/proof the equivalence of the 2 sets of 3 equations.

for instance, from starting point 1, I obtain the following 3 equations

$\displaystyle \\p_1 = \frac{1}{b_1} + a_1 = 3 \frac{c^3+1}{c^3-1} \\ p_2 = \frac{a_1}{b_1} + a_2 = 3 \\ p_3 = \frac{a_2}{b_1} \qquad = \frac{c^3+1}{c^3-1}$

and from starting point 2, I obtain the following 3 equations

$\displaystyle \\ 2(1+q_1+q_2+q_3) &=& c(q_1-q_2-3(q_3-1)) \\ \qquad(1+q_1+q_2+q_3) &=& c^3(q_2-q_1+1-q_3) \\ 2(1+q_1+q_2+q_3) &=& c^2(3(q_3+1)-q_2-q_1)$

where

$\displaystyle \\q_1 = a_1+b_1 \\ q_2 = a_2+a_1b_1 \\ q_3 = a_2b_1$

The two sets of 3 equations are equivalent in the sense that they lead to the same solution for the 3 unknows $\displaystyle a_1, a_2, b_1$ given any $\displaystyle c$, I have verified this numerically.

My problem is that I'm having problems showing this. At first sight it seems that it might be easy to show this, however, after having spend considerable time on this I have started to worry that it might not be so easy.

Any comments or feedback on this problem is highly appreciated.