# How to solve this differential eq problem?

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?

### 1 Answer

- rotchmLv 79 months agoBest Answer
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.

Hi! We have not learned to integrate in my class- is there some way to solve without integration? Thank you!