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To solve the differential equation \( \frac{dP}{dt} = (k \cos t) P \), the method used is known as separation of variables. This technique is particularly useful for first-order differential equations and involves rearranging the equation to separate the differentials of each variable on opposite sides. Here, we start by dividing both sides by \( P \) and multiplying by \( dt \), leading us to \( \frac{dP}{P} = k \cos t \, dt \).
This process effectively isolates all terms involving \( P \) on one side and all terms involving \( t \) on the other. By doing this, we create an opportunity to integrate each side with respect to its own variable.
The main benefit of separation of variables is that it simplifies the integration process and can be used in various other types of first-order differential equations as well.

After separating the variables, the next step is to integrate both sides of \( \int \frac{dP}{P} = \int k \cos t \, dt \). Integration involves finding the antiderivative of a function. For the left side, the integral of \( \frac{1}{P} \) with respect to \( P \) is \( \ln |P| \).
On the right side, the integral of \( k \cos t \) with respect to \( t \) is \( k \sin t + C \), where \( C \) is the constant of integration. This constant is added because indefinite integration does not specify the bounds, so any number could potentially be added to the function.

• Left side integral: \( \int \frac{dP}{P} = \ln |P| \)
• Right side integral: \( \int k \cos t \, dt = k \sin t + C \)

Integration is a crucial step as it allows us to find a function that describes the behavior of the system represented by the differential equation.

The given differential equation \( \frac{dP}{dt} = (k \cos t) P \) serves as a mathematical model for a population \( P(t) \). Mathematical modeling involves using equations to represent real-world phenomena.
Here, the model suggests that the rate of population change is influenced by a constant \( k \) and a sinusoidal function \( \cos t \). This implies that the population experiences periodic fluctuations over time, similar to seasonal changes.

• \( k \) is a positive constant affecting the amplitude of fluctuations.
• \( \cos t \) indicates the periodic nature of the changes.

This model is valuable as it provides insights into patterns and predictions for population dynamics subject to cyclical variations.

An essential aspect of solving differential equations is applying initial conditions to determine the specific solution. The initial condition given is \( P(0) = P_0 \). This means at time \( t = 0 \), the population is \( P_0 \).
Substituting \( t = 0 \) and \( P = P_0 \) into the integrated equation \( \ln |P| = k \sin t + C \), gives us \( \ln |P_0| = C \). Thus, \( C = \ln P_0 \).
Applying initial conditions is critical as it allows us to solve for unknown constants and arrive at a unique solution that fits the specific circumstances of the problem. With \( C \) known, the particular solution becomes \( P(t) = P_0 e^{k \sin t} \), which can now be used to predict future behavior under the initial conditions specified.