Hey all, i posted this over at reddit math help but they didn't respond and the one responder said i made a mistake but i'm not sure what it is? I haven't integrated in forever and he said i messed up my integration, so i'm turning to you guys with hopes you'll explain it

The relevant parts of the question are:

Two companies produce a product at $3 per. There are no other costs, and the two companies are trying to maximize profits (Y).

Q1=12-P1+.5P2

Q2=12-P2+.5P1

Y1=Q2(P1-3)

Y2=Q1(P2-3)

A) Find the equilibrium price and profit for each firm.

Here is what i have:

π1=TR-TC

π1=P1Q1-3Q1

π1=P1(12-P1+.5P2)-3(12-P1+.5P2)

π1=15P1-P1^{2}+.5P1P2-1.5P2-36

dπ1/dP1=15-2P1-P2

15-2P1-P2=0

-2P1=P2-15

P1=7.5-.5P2 (Reaction curve for firm 1 and 2)

Substitute P2 into P1 to find equilibrium price

P1=7.5-.5(7.5-.5P1)

P1=3.75+.25P

*P1=5 (equilibrium price)

Plug P into demand function

Q1=Q2=12-5+.5(5)=9.5

Put P and Q into profit equation

Y1=Y2=Q(P-3)

*Y=9.5x2=19 (in thousands)

So i think there's a mistake with my work because i'm having an issue with another section later.

B) If the firms collude and choose a joint best price P, the profit for each becomes Y1-Y2=(P-3)(12-P+.5P). What price maximizes their joint profit?

π=PQ-3Q

π=P(12-P+.5P)-3(12-P+.5P)

π=12P-.5P^{2}-36+3P-1.5P

π=13.5P-.5P^{2}-36

dπ/dP=13.5-P

*P=13.5 (profit maximizing price)

C) Suppose one firm chooses to defect while the other maintains the agreed price, what is the best defecting price? What is the resulting profit? (here is where i run into issues)

Plug 13.5 into original reaction curve

P2=7.5-.5(13.5)

P2=.75

This is clearly wrong, because cost per product is $3, and this induces massively negative profit. Furthermore, i know that the right answer is 13.49, and my professor told me that there must be an error in one of my calculations (i emailed him). I knew the answer would be 13.49 but i still can't get it with the math (a detail of the problem is that both companies have loyal clients and floaters who will go to the firm with the lower price, so undercutting by 1 cent is the maximizing strategy).

Does anyone see where i am going wrong?