Simple Differiential equation, just finding a constant C problem .:Resolved:.

[voidflux]

Hello everyone, I'm close to the solution but i just got hung up on some sort of algebra mistake i think.....

I'm solving a Logistics Equation, dealing with population growth/decay
All i have to do is use P(0) = Po to get C2 and then it will be solved.
Po stands for P sub o. not P*0.

I have

P/(M-P) = C2 e^(mkt)

note: C2 is just a constant C, u get when u integrate.
M and k are constants.

I now the solution is:

P(t) = MPo/[Po + (M-Po)*e^(-kmt)]

Here is what I did,

P/(M-P) = C2*e^(mkt)
P = C2*e^(mkt) (M-P)
now I used P(0) = Po;
Po = C2*e^(mk(0)) (M-P)
C2 = Po/(M-P);
Then i plugged C2 back into the orginal equation and I got
P/(M-P) = [Po/(M-P)]*e^(mkt);

which looks nothing like the solution...any ideas where I messed up?

Thanks for listening!

[krtxmrtz]

To avoid excessive overhead let's leave P(0) aside for now.

P/(M - P) = C₂e^mkt

Invert:

(M - P)/P = 1/(C₂e^mkt)

Perform operations on left hand side:

M/P - 1 = 1/(C₂e^mkt)

M/P = 1 + 1/(C₂e^mkt)

Invert again:

P/M = 1 / ( 1 + 1/(C₂e^mkt)) = 1/(1 + e^-mkt/C₂)

P = M / (1 + e^-mkt/C₂)

OK, now when t = 0 you have P(0) = P₀, therefore:

P₀ / (M - P₀) = C₂

So that:

P = M / (1 + e^-mkt(M - P₀)/P₀) = MP₀ / (P₀ + (M - P₀)e^-mkt)

[voidflux]

Excellent job, thanks alot!!

[krtxmrtz]

Originally posted by voidflux
Excellent job, thanks alot!!

You're welcome