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[apcalculus] Re: is this separable?
Posted:
Mar 3, 2005 9:30 AM


Hello Gay,
On Wed, 2 Mar 2005, Gay L Lawrimore wrote:
1. Do you think AB will have to solve one like this and 2. I still would like to know how to do this.
dP/dt = 2P(3P) Show that P(t) = 3/(1+2e ^(6t)) is a solution. In previous parts they say sketch a solution which passes through (0,4) and suppose P(t) = 1, but not in this part.
I'm not sure about 1. but as far as 2. goes, write
dP/(P(P3)) = 2 dt
Now, 1/(P(P3)) can be written as
(1/3)/P + (1/3)/(P3)
{you can use algebra, partial fractions, or just fooling around} so that the left hand side then integrates to
ln(abs((P3)/P)))
and the right hand side integrates to
6t+C*
where C* is your constant of integration. If you exponentiate both sides you then get  after combining a couple of steps 
(P3)/P = Ae^(6t)
where A is an arbitrary constant. Do some algebra and you get
P = 3/(1Ae(6t))
I'm not sure what the (0,4) has to do with anything  I don't have the text you mentioned  but I do know that if we require P(0)=1, then A = 2 and we get the solution you mentioned.
Hope this helps.
Dick Maher
Richard J. Maher Mathematics and Statistics Loyola University Chicago 6525 N. Sheridan Rd. Chicago, Illinois 60626 17735083565 rjm@math.luc.edu
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