We are given the differential equation:

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

To solve this equation, we'll follow your instructions and perform the necessary transformations step-by-step.

Step 1: Rearrange the equation

We begin by isolating terms involving $A$ on the left-hand side and terms involving $t$ on the right-hand side.

Starting with the equation:

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

We can rearrange this as:

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

Now, we have separated variables with all terms involving $A$ on the left and those involving $t$ on the right.

Step 2: Integrate both sides

We now need to integrate both sides with respect to their respective variables.

• Left side: The left-hand side involves $A$, so we need to integrate:

$\int \frac{1}{\sqrt{A} \left( 1 - \frac{A}{M} \right)} \, dA$

• Right side: The right-hand side is a straightforward integral:

$\int k \, dt = kt + C$

Now, let’s focus on integrating the left side.

Step 3: Substitution to simplify the integral

The integral on the left side looks tricky, so we'll use a substitution to simplify it. Let's consider the substitution:

$u = \sqrt{A}, \quad \text{so} \quad A = u^2$

Differentiating $A = u^2$ with respect to $t$ gives:

$\frac{dA}{dt} = 2u \frac{du}{dt}$

Now, we need to rewrite the left-hand side integral using this substitution. Using $A = u^2$, we can rewrite the expression:

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

Thus, the left-hand side integral becomes:

$\int \frac{1}{u \left( 1 - \frac{u^2}{M} \right)} \, du$

This integral will require a standard method, such as partial fraction decomposition or another substitution, to simplify further. However, the key step here is transforming the equation into this solvable form.

Step 4: Solution and simplification

After integrating, we will end up with an expression involving $u$, which can be substituted back into $A = u^2$. We will then solve for $A(t)$.

Next Steps

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