Solution to 1st order nonlinear differential equation

[gulruozkan]

Hi,
I'm a PhD student in Operations Management, and I've stumbled across a differential equation while modeling an OM concept. I was wondering if you could help me with this differential equation, or direct me in a way that would help me solve it.

The equation is:
y'[t]+A[t]*(y[t])^2+B[t]*y[t]+G[t]=0.

As you can see, y, A, B, G are all functions of t. Unfortunately, due to nature of the functions A, B and G, I cannot transform the above equation into y'[t]+(y+J[t])^2=0, which would help me use substitution and solve the above equation. If I can get y[t] solution as a function of A[t}, B[t] and G[t], it would really help me with my research.

Thanks a lot for your help in advance.
Best,

Gulru

[lzkelley]

I'm by no means an expert of DE's, but the format of your equation looks like it might lend itself to a power series substitution method - it would be a certain amount of brute force, but since you're interested in applications it might suffice.

[Mute]

I'm by no means an expert of DE's, but the format of your equation looks like it might lend itself to a power series substitution method - it would be a certain amount of brute force, but since you're interested in applications it might suffice.

The nonlinearity in the problem makes a power series solution very difficult, since you have to multiply series together for the y^2 term. It would really only be plausible if you were sure you could neglect terms and retained only a few terms in the series.

Anywho, fortunately you probably don't quite need to resort to that yet, as this equation is of the form of a Riccati equation:

http://en.wikipedia.org/wiki/Riccati_equation