How to convert DFA to regular expression using arden's rule

Question

I generally use state removal method to convert DFA to regular expressions. But I want to try other methods like using algebraic methods and arden's rule.

How to solve a DFA with multiple final states using algebraic method? I refered this link https://cs.stackexchange.com/questions/2016/how-to-convert-finite-automata-to-regular-expressions and arrived at this solution which I think is wrong since a can be followed by only a or c not b.

DFA

solution

I used state removal method and arrived at below regular expression which seems to be correct as the language won't contain ab.

state removal method

Answer 1

Your equations are wrong; your solution works for the reversed arrows. You should be solving the following system: $$ q_0 = (b+c)q_0 + aq_1 + \epsilon \\ q_1 = cq_0 + aq_1 + \epsilon $$