AFAIK sequential circuits do not have characteristic equations. Their behavior is not governed by differential equations.
I can tell you that the device has four states with AB = {00, 01, 10, 11}, and one output Y, which is (X⊕A)⊕B.

 

The notion of "characteristic equation" if very context dependent.

https://en.wikipedia.org/wiki/Excitation_table

The characteristic equation of a sequential circuit is simply the equation(s) that gives the next value(s) of the state variable(s) in terms of the input variable(s) and the current value(s) of the state variable(s).

Show your best attempt and then we can see if you are doing it right and help you along if you aren't.

 

S1 = B and R1 = B’.
S2 = (X⊕A)’ and R2 = (X⊕A).as
Q+ = S + QR’
A next = B + AB” = B + AB.


B next = (X⊕A)’ + B (X⊕A)’.
Y = (X⊕A) ⊕B

can you please tell how close i am?

 

can you please tell how close i am?

If the present state is (X=1 A=1 B=1 Y=1) then wouldn't the next state be (X=1 A+=1 B+=1 Y+=1)?

 

S1 = B and R1 = B’.
S2 = (X⊕A)’ and R2 = (X⊕A).as
Q+ = S + QR’
A next = B + AB” = B + AB.


B next = (X⊕A)’ + B (X⊕A)’.
Y = (X⊕A) ⊕B
View attachment 124866
can you please tell how close i am?

The sub-columns in your Present State column need labels -- don't make readers (including the grader) guess what you mean. Engineering is not about guessing.

Can you see how you can reduce the expression for Bnext? Does this simplification make sense?

Do your next state columns agree with your equations?

Let's say that A=0, B=1, X=0. What is (X⊕A)? What should Bnext be? What does your table show it to be?

We aren't doing anything to check your work that you shouldn't be able to do to check your work. You really need to develop the ability to check the validity, including the self-consistency, of your own results.

 

Since this state machine has two state variables (A & B) and one control input (X), there are eight possible input combinations which determine the next state. I believe the simplest method to obtain the complete next-state table is to build the machine and log its performance. So that is what I did, but added buttons to the preset and clear inputs of each SR flip-flop so it was easy to set the state of the machine, then tick the clock and record the next-state.

Note that the simulator I prefer is Logisim. It is free software and runs on Linux (and others) as a Java JAR file. Logisim also has a combinational analytics capability where it will take the state table, draw the Karnaugh maps, and provide the simplified characteristic equations for the next-state variables.