How to Find S as a Function of t in Differential Equations?

[olechka722]

Hi, it has been a little while since I have actually had a course in this, and my math memory is terrible. I would like to find S as a function of t, when the differential relationship is the following:

dS/dt= A- (B*S/(S+C))*(D-E*t) where all of the other letters are just constants.

What would be the correct approach here? Is it just numerical integration?

 

[HallsofIvy]

If you mean just integrate both sides, no, it's not. You are looking for the unknown function S and you have S on the right side as well so you can't just integrate with respect to t.

That is, rather a differential equation.

If A were equal to 0, it would be "separable". We could "separate" the variables S and t as
$$\frac{S+C}{S}\frac{dS}{dt}= -B(D- Et)$$
and you can integrate both sides of that with respect to s:
$$\int\frac{S+C}{S}\frac{dS}{dt}dt= \int(-BD+ BEt)dt$$
$$\int\frac{S+C}{S}dS= \int(-BD+ BEt)dt$$.

However, with that "A", it's not that simple. There are a variety of ways of solving first order differential equation but I don't see any immediately that would work for that equation.

 

[olechka722]

That is pretty much what I thought. Thank you!

I am going to just do numerical integration.