So, I have to solve for the following set of equations

$ q_1$ = $ q_1$ a + $ q_2$ b + $ \epsilon$

$ q_2$ = $ q_1$ a + $ q_2$ b + $ q_3$ a

$ q_3$ = $ q_2$ a

There are two ways to do this

I did this

$ q_1$ = $ q_2$ b + $ \epsilon$ + $ q_1$ a

$ q_1$ = ($ q_2$ b + $ \epsilon$ )a* Applying ardens theorem

Now substituting in $ q_2$ the values of $ q_1$ and $ q_3$

$ q_2$ = $ q_2$ ba*a + a*a + $ q_2$ b + $ q_2$ aa

$ q_2$ = a*a + $ q_2$ (ba*a+b+aa)

$ q_2$ = a*a(ba*a+b+aa)* Applying ardens theorem

$ q_2$ = a*a(ba*+aa)*

Now substituting in $ q_3$ , the answer should be

$ q_3$ = a*a(ba*+aa)*a

However, the correct answer is

(a + a(b+aa)*b)*a(b+aa)*a

which can be obtained by first substituting $ q_3$ in $ q_2$ , and then substituting $ q_2$ in $ q_1$ , and finally solving $ q_2$ , $ q_3$ from the obtained regular expression for $ q_1$ .

Can someone tell where I have gone wrong in the above method, or am I applying ardens theorem in the wrong way ?