Final answer:
To solve the differential equation dp/dt = kP - 0.1P², we apply separation of variables and integrate both sides. The general solution involves partial fraction decomposition and leads to a logarithmic function, resulting in the population formula P(t) = k/(0.1 + Ce^{-kt}) with C as the constant of integration.
Step-by-step explanation:
To find the general solution of the differential equation dp/dt = kP - 0.1P², we can use the method of separation of variables. This method involves rearranging the equation so that all terms involving P are on one side and all terms involving t are on the other side. The separated equation can then be integrated with respect to P and t respectively.
The first step is to separate the variables:
\(\frac{1}{P(k - 0.1P)}\,dp = dt\)
Next, we integrate both sides:
\(\int \frac{1}{P(k - 0.1P)}\,dp = \int 1\,dt\)
This integral can be tricky but it can be solved by partial fraction decomposition. Assuming that k is not equal to zero, we can rewrite the left-hand side as two separate fractions:
\(\frac{A}{P} + \frac{B}{k - 0.1P}\)
After finding A and B by equating coefficients, the integral becomes a sum of two simpler integrals, typically resulting in logarithmic functions. After integrating and applying the constant of integration, we obtain an implicit solution for P(t).
The result is:
\(P(t) = \frac{k}{0.1 + Ce^{-kt}}\)
where C is the constant of integration, which can be determined by initial condition if provided.
The solution represents the population P(t) as a function of time t, and it incorporates the effect of both linear growth and a quadratic term that represents limitations such as resource competition affecting the population.