Generally Ricatti equations can be useful in solving nonlinear recurrence relations; however, I don't think they apply to your problem. I had a similar problem to solve for my thesis. I solved it as follows:

This is a first order, non-linear difference equation with variable coefficients. Commonly used solution methods such as Ricatti Equations do not seem to work nicely for this example. However, by inspection, it seems that the solution is a linear function . From the definition, . This gives that ; hence, . Thus we have . Substituting this into the difference equation, we get

Furthermore, the above holds for all , so we can choose an to solve for . Take to get

Taking would yield negative solutions which is not possible; hence, and . Thus we have that is a particular solution to the difference equation. Hence, we get