Second Order Difference Equation- Cournot

Hi , i hope someone can help me with this. I am aware of how to solve diffrence equations however unsure about how to approach this. any help greatly appreciated.

I have part of the answer showing partial differentiation but i dont see how thats applicable to this....

2 firms supplying the market with demand function

P(x+y)= alpha - beta(x + y)

x and y are output levels of Firm1 and Firm2 respectively. assume each firm makes Cournot conjecture and find second order difference equation

(cournot conjecture implies each firm chooses its output level fot <t+1> to max profit in belief that the other firm will maintain output at period <t> value)

This is the solution im given, there are however many steps missing and im wondering if anyone could explain the solution to me especially obtaining the 1st line and reason for partial differentiation.

Ot+1 = <alpha -beta(Xt+1 + Yt)> Xt+1

dOt+1 = alpha- 2betaXt - betaYt

dXt

Xt+1 = alpha/2beta - 0.5Yt

Ot+1= <alpha - beta(Xt + Yt+1)> Yt+1

dOt+1 = alpha - 2betaYt+1 - betaXt = 0

dYt

So Yt= alpha/2beta - 0.5 Xt-1

So Xt - 0.25Xt-2 = 0