(You may want to maximize your window to see the solution more clearly.)
We want to verify that
is a solution of the differential equation. But first, we will simplify the notation:
Let 
Therefore,
Now find the derivative with respect to t,
Thus,
But we also have,
Simplifying and combining terms with a common denominator gives,
And finally,
which is the same as the right-hand side of equation (1); thus,
solves the given differential equation.
To examine the asymptotic behavior, we take the limit as time approaches infinity,
Because M and P0 are constants, only the exponential will be affected by the limit. Because r (the growth constant) is positive, then
Thus,
That is as expected (because M is the limiting population value that can be supported by assumption), the population is asymptotic to y = M.