The expression you’ve provided is a differential equation:

$\frac{d(p)}{d(t)} = -\frac{2kpm}{1 + mkt}$

This equation involves the rate of change of $p$ with respect to $t$, and it contains the parameters $k$, $m$, and $p$.

To better understand the behavior of $p(t)$, we can attempt to solve this equation.

Step 1: Separation of Variables

Rearrange the equation to separate the variables $p$ and $t$:

$\frac{1}{p} \, dp = -\frac{2km}{1 + mkt} \, dt$

Step 2: Integrating Both Sides

Now, we can integrate both sides. We will treat $m$, $k$, and $p$ as constants.

Left-hand side:

$\int \frac{1}{p} \, dp = \ln |p|$

Right-hand side:

The right-hand side requires a bit more attention. Let's focus on the integral:

$\int \frac{2km}{1 + mkt} \, dt$

Let’s make a substitution to simplify this integral. Set:

$u = 1 + mkt \quad \text{so that} \quad du = mkt \, dt$

Thus, the integral becomes:

$\int \frac{2km}{u} \, du = 2k \ln |u| = 2k \ln |1 + mkt|$

Step 3: Final Solution

After integrating both sides, we get the general solution:

$\ln |p| = -2k \ln |1 + mkt| + C$

Where $C$ is the constant of integration. Exponentiating both sides:

$|p| = e^C (1 + mkt)^{-2k}$

Let $e^C = A$ (where $A$ is a constant), so the solution simplifies to:

$p(t) = A (1 + mkt)^{-2k}$

Conclusion:

The solution to the differential equation is:

$p(t) = A (1 + mkt)^{-2k}$

Where $A$ is a constant determined by initial conditions.

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Would you like a more detailed explanation or have any other questions about this solution? Here are some related questions for further exploration:

1. How would you find the constant $A$ given an initial condition, such as $p(t_0) = p_0$?
2. How does the value of $k$ affect the behavior of $p(t)$ as $t \to \infty$?
3. What happens to the solution as $t \to 0$?
4. How would you modify the equation if $m$ and $k$ were time-dependent functions?
5. How would you handle cases where $p(t)$ becomes negative?

Tip: Remember that exponential growth or decay behavior is influenced by the form of the equation, such as $(1 + mkt)^{-2k}$ showing a decaying behavior for $k > 0$.