Why Does This Differential Equation Epidemic Model Stump Me?

[Kudaros]

I have a logistic issue with differential equations. I have spent four hours working on this problem and it is way past the point of ridiculous.

If anyone can help out that would be great.

It begins like this.

dy/dt = k(P-y)*y. It is an epidemic model, where k is a positive constant relating the rate of infection. P is total population involved. An initial condition of y(0)=t is given. I am to find this particular solution.

The y is multiplied. Strange that it isn't shown as ky(p-y) but perhaps its a hint that I am not getting.

Anyway, I am treating this as a separable equation as that is the tool we are given (the method).

I have tried numerous approaches and have gotten the farthest with this one.

dy/(y(p-y)= kdt

Partial fractions on the left side where A and B are both = to 1/p.

1/py + 1/(p(p-y)) dy = kdt

Then integrate both sides.

ln(y)/p - ln(p-y)/p = kt + c.

Here is where the problem begins (unless I screwed up earlier). A CAS gives me -ln(y-p) (second term left side) . Can anyone explain this? I integrated by U substitution. Why would the variable and constant be switched?

I continued on assuming the CAS was correct and now I cannot isolate Y variable. I have never had this trouble before with math but hey I guess that's the way it goes.

Any help would be greatly appreciated.

 

[teclo]

trust your math skills, not mathematica! (or whatever program you're using). the difference arises if you factor a negative one out before integrating. thus, you are then integrating - 1/p int (1/y-p) using u substitution now requires no extra negative for du. make sense?

 

[arildno]

Remember that:
[tex]\int\frac{dx}{x}=\ln(|x|)+C[/itex][/tex]

 

[Kudaros]

p-y also makes logical sense, as y is the number of infected persons and cannot be greater than p, ensuring a nonnegative number.

Continuing along the problem however, it seems I cannot isolate the variable Y due to the term p-y. I have attempted numerous approaches finding that same problem. Is there another way to solve differential equations, in general, using the initial condition but without isolating Y?

More specific to this problem, are there any assumptions that can be made about p to eliminate it all together? (a longshot I realize.)

Thanks again!