Second-order recurrence equations

2nd order recurrence equation is when the equation is of the form

yt + Cyt-1 + Dyt-2 = k with t > 2 and yt+2 + Cyt+1 + Dyt = f(t) with t > 0

It goes back 2 time periods. B and C are coefficients in a linear second-order recurrence equation with constant coefficients.

y0 and y1 or some variant will be given in the question.

Substitue yt, yt-1 and yt-2 with k(or some other letter like gamma) to make it a quadratic equation

The real solution of yt is of the form ~> yt = Ak1^t + Bk2^t

Make the highest t have the highest power eg: k^2 + Bk + C = 0

Solve for k then substitute the values of k into the yt general solution form

since y0 and/or y1 or other values may be given, solve for them to find coefficients A and B

Substitute A and B into the final solution

A repeated solution for when k is substituted will take the form yt = (At + B)kt

If there is no real solution, the solution will be complex numbers we can use to solve the recurrence equation.

The complex numbers in the solution can be re-written as real solutions however.....