A population tries to grow exponentially but is limited by the resources of its environment. This growth is modeled by the differential equation \[ \frac{d P}{d t}=\frac{k}{M} P(M-P) \] Find a solution for this differential equation using the continuous growth rate $k=0.4$, the carrying capacity $M=20$, and the initial population size $P(0)=3$. \[ P(t)= \] Hint: You can solve the differential equation using separation of variables, but the calculations may be easier if you treat it as a Bernoulli equation.

Step 1 ：Given the differential equation \(\frac{d P}{d t} = \frac{k}{M} P(M-P)\), where the continuous growth rate \(k=0.4\), the carrying capacity \(M=20\), and the initial population size \(P(0)=3\).

Step 2 ：Separate the variables P and t on different sides of the equation.

Step 3 ：Integrate both sides of the equation to find the solution. The solution to the differential equation is \(P(t) = -\frac{20.0}{C1e^{-0.4t} - 1.0}\).

Step 4 ：Use the initial condition \(P(0) = 3\) to find the constant of integration. The equation becomes \(3 = -\frac{20.0}{C1 - 1.0}\).

Step 5 ：Solve the equation to find the value of \(C1\), which is \(-5.66666666666667\).

Step 6 ：Substitute \(C1\) into the solution, the final solution to the differential equation is \(P(t) = \frac{-20.0}{-1.0 - 5.66666666666667e^{-0.4t}}\).

Step 7 ：\(\boxed{P(t) = \frac{-20.0}{-1.0 - 5.66666666666667e^{-0.4t}}}\) is the final answer.