How to solve this differential eq problem?

Question

Consider the diff eq: (dP/dt) = kP(L-P).

K an dL are positive costants. How can you show that P(t) = L/(1 + Ce^(-kLt)), where C is constant, is a solution of this logistic equation?

Answer 1

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You can show it by differentiating the given P(t) and showing that it equals the LHS.

Or you can find the solution by solving the DE this way: From highschool algebra, rewrite the DE as

(1/L) * ( 1/P + 1/(L-P) ) dP = k dt.

Now the calculuse part: integrate both sides. Then once more a little high school algebra to isolate P and The answer follows.

Done!

To your comment. yes there is, and I told you how in my first line!

Again, you are given (dP/dt) = kP(L-P).

And u are given explicitly P(t). So differentiate it (the LHS). And for the RHS, just replace the given P(t).

A little algebra shows that the LHS & RHS are the same.