Question

Verify that P = Ce^t /1 + Ce^t is a one-parameter family of solutions to the differential equation dP dt = P(1 − P).

Answer 1

See verification below

Explanation:

We can differentiate P(t) respect to t with usual rules (quotient, exponential, and sum) and rearrange the result. First, note that

`1-P=1-(ce^t)/(1+ce^t)=(1+ce^t-ce^t)/(1+ce^t)=(1)/(1+ce^t)`

Now, differentiate to obtain

`(dP)/(dt)=((ce^t)/(1+ce^t))'=((ce^t)'(1+ce^t)-(ce^t)(1+ce^t)')/((1+ce^t)^2)`

`=((ce^t)(1+ce^t)-(ce^t)(ce^t))/((1+ce^t)^2)=(ce^t+ce^(2t)-ce^(2t))/((1+ce^t)^2)=(ce^t)/((1+ce^t)^2)`

To obtain the required form, extract a factor in both the numerator and denominator:

`(dP)/(dt)=(ce^t)/(1+ce^t)(1)/(1+ce^t)=P(1-P)`