The Logistic Equation
I was wondering if anyone could help me with my logistic equation problem; the question asks me what logistic equation is solved by y=2/(1+e^-t).
If I take the derivative from above I get 2et/(e^t+1)^2, however I have no idea how to get it to the form y=cy-by^2. I know it has something to do with finding the residue and splitting the variables, but I am stuck.
You can solve the differential equation y' = cy - by^2, and write the solution in terms of generic c and b.
Then see how you should choose c and b to get y=2/(1+e^-t).
I solved y'=cy-by^2 to get y=c/(b+(1/C)e^(-ct)) but I'm not sure how I'm supposed to solve this for the correct logistic equation.
What values should you pick for c and b to get the solution into the form you are looking for 2/(1+e^-t)?
Okay, I've rearanged it so that Ce^ct=2/(c(1+e^-t)-2b)) but I don't know where to go from there. I think I have to take the ln() of the left side and multiply it by 1/c, but I'm not sure where to go from there. Sorry I'm so slow, but this is strange as this question is done backwards from what we were taught.
The key thing is to notice for
c/(b+(1/C)e^(-ct)) = 2/(1+e^-t)
We must have e^(-ct) = e^-t. What is c?
Now can you solve for b?