Hi, I am supposed to solve the following initial value problem but am stuck with an ugly integral:
y' = y(1-y) y(0)=yo (y subscript 0), where is an arbitrary constant.
I have worked it out as follows:
dy/dt = (y-y)
dy/(y-y) = dt
[ dy/(y-y) ] = [ dt ]
[ dy/(y-y) ] = t + c
But how do you integrate the left side? Or is there another way to solve this for y? Thank you.
I seem to have run into another snag. I solved the integral and have: ln|y|-ln|y-1|=t+c. Solving for y, I obtained:
y= -e^(t+c) / (1-e^(t+c))
however, the answer that I am supposed to find is y=yo/[yo+(1-yoe^-t)],
where y(0) = yo. How did this result get obtained from the integral??
Err, for some reason the y and yo are confusing me. From this point, I did:
1 / (y-1) = -1+Ae^t
y-1 = 1 / (-1+Ae^t)
y = (1 + Ae^t - 1) / (Ae^t - 1)
y = (Ae^t ) / (Ae^t - 1)
when t=0, y=yo
yo = A/(A-1)
yoA-yo = A
-yo = A(1-yo)
A = (1-yo)/-yo = (yo-1)/yo
And now I am stuck again. When I plug this in for A, I still cannot get it to look right..