We are given the differential equation:

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

Also, you’ve provided the integral:

$\int \frac{1}{A\left(1 - \frac{A^2}{M}\right)} dA = -\frac{1}{2} \left( -2\ln(A) + \ln|M| + \ln\left|\frac{A^2}{M} - 1\right| \right) + C$

This equation suggests the form of the solution involves integrating the differential equation and then solving for $A(t)$.

Step 1: Rewrite the Integral

First, we simplify the integral result to help solve for $A(t)$.

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

Step 2: Solve for $A(t)$

The general solution to this form of equation is given by:

$\ln(A) - \frac{1}{2} \ln\left|\frac{A^2}{M} - 1\right| = kt + C_1$

You can rearrange this to isolate $A(t)$ by solving this equation further. However, the next step involves the given relationship $B = \sqrt{A}$, which simplifies the process.

Step 3: Express $B(t)$

Since $B = \sqrt{A}$, we can square both sides to relate $B$ and $A$:

$A = B^2$

Thus, we will substitute $A = B^2$ into the equation derived above for $A(t)$, and solve for $B(t)$.

Would you like me to continue solving this equation for $B(t)$ explicitly or clarify any part?

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Here are five related questions you might find useful:

1. How can we simplify the equation for $A(t)$ further?
2. What would the solution look like if we had a specific initial condition for $A(0)$?
3. How does the behavior of $A(t)$ change as $t$ increases?
4. Can we derive the solution for $A(t)$ using a different method like separation of variables?
5. What happens to $B(t)$ as $t$ approaches infinity?

Tip: When dealing with nonlinear differential equations like this, it can help to introduce auxiliary variables (like $B = \sqrt{A}$) to simplify the expressions and obtain a clearer solution.