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Unformatted text preview: -t 1 ) . Solution: As for the logistic equation, we can proceed as usual: dp dt = ap-bp 2 Z dp ap-bp 2 = Z dt Z 1 /a p dp + Z b/a a-bp dp = t + C 1 /a ln | p | -1 /a ln | a-bp | = t + C 1 a ln p a-bp = t + C only care about p > p a-bp = aCe at p = ( a-bp ) aCe at p = a 2 Ce at 1-baCe at p ( t ) = a/b 1 + e Ce-at Now we can determine the remaining constant e C by means of the initial condition, as clearly the limiting population is a b : 1 2 a b = p ( t 1 ) = a/b 1 + e Ce-at 1 solving this gives 2 = 1 + e Ce-at 1 and hence we have e C = e at 1 Substituting this back in to the general solution gives p ( t ) = a/b 1 + e-a ( t-t 1 )...
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