Show directly that P is a solution to the differential equation

[phosgene]

Homework Statement

Show directly that
P=\frac{L}{1+Ae^{-kt}}

is a solution to the differential equation

dP/dt=kP(1-P/L)

Homework Equations

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The Attempt at a Solution

I assume that all I need to do is differentiate P with respect to t. However, as you can see below, either I'm doing it wrong or I don't know what I need to do next.

dP/dt=(d/dt)\frac{L}{1+Ae^{-kt}}

First I take 1+Ae^-kt to be 1/x, and differentiate it to -x^-2. Then I multiply it by the derivative of x, which is -Ake^-kt. So I finish up with \frac{LAe^{-kt}}{(1+Ae^{-kt})^{2}}

Im stuck here. I have no idea how to get the correct answer :S

 

[SammyS]

Homework Statement

Show directly that
P=\frac{L}{1+Ae^{-kt}}

is a solution to the differential equation

dP/dt=kP(1-P/L)

Homework Equations

The Attempt at a Solution

I assume that all I need to do is differentiate P with respect to t. However, as you can see below, either I'm doing it wrong or I don't know what I need to do next.

dP/dt=(d/dt)\frac{L}{1+Ae^{-kt}}

First I take 1+Ae^-kt to be 1/x, and differentiate it to -x^-2. Then I multiply it by the derivative of x, which is -Ake^-kt. So I finish up with \frac{LAe^{-kt}}{(1+Ae^{-kt})^{2}}

I'm stuck here. I have no idea how to get the correct answer :S

You're missing k in that last expression:

\displaystyle \frac{kLAe^{-kt}}{(1+Ae^{-kt})^{2}}​

Multiply the numerator & denominator by L .

Change the Ae^{-kt} in the numerator to \left(1+Ae^{-kt}\right)-1

 

[phosgene]

Thanks! I'll try it out and see if I can get it.