The differential equation you’ve provided is:

$\frac{dA}{dt} = kA\left(1 - \frac{A^2}{M}\right)$

This is a separable differential equation. We can proceed by separating the variables and solving.

Step 1: Separate the variables

Rearrange the equation to isolate terms involving $A$ on one side and $dt$ on the other side:

$\frac{dA}{A\left(1 - \frac{A^2}{M}\right)} = k \, dt$

Step 2: Simplify the left-hand side

The left-hand side can be simplified by factoring:

$\frac{1}{A\left(1 - \frac{A^2}{M}\right)} = \frac{1}{A\left(\frac{M - A^2}{M}\right)} = \frac{M}{A(M - A^2)}$

Thus, the equation becomes:

$\frac{M}{A(M - A^2)} \, dA = k \, dt$

Step 3: Perform partial fraction decomposition

To integrate the left-hand side, we use partial fraction decomposition. The expression $\frac{M}{A(M - A^2)}$ can be rewritten as:

$\frac{M}{A(M - A^2)} = \frac{1}{2A} \left(\frac{1}{M - A^2}\right)$

This simplifies to:

$\frac{1}{M - A^2} \left( \frac{1}{A}\right)$

Integrating both sides gives the full form