52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.


where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.


a. Verify by substitution that the general solution of the equation is P(t) = K/(1 + Ce⁻ʳᵗ), where C is an arbitrary constant.

Verified step by step guidance

1. Start with the given differential equation, which is typically the logistic growth model: \[\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right),\] where \(P(t)\) is the population at time \(t\), \(r > 0\) is the growth rate, and \(K > 0\) is the carrying capacity.
2. Write down the proposed general solution: \[P(t) = \frac{K}{1 + Ce^{-rt}},\] where \(C\) is an arbitrary constant.
3. Compute the derivative of \(P(t)\) with respect to \(t\). Use the quotient rule or rewrite \(P(t)\) as \(K(1 + Ce^{-rt})^{-1}\) and apply the chain rule to find \[\frac{dP}{dt}.\]
4. Substitute \(P(t)\) and \(\frac{dP}{dt}\) back into the original differential equation \[\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)\] and simplify both sides to check if they are equal.
5. If both sides are equal after simplification, this verifies that the proposed \(P(t)\) satisfies the differential equation, confirming it is indeed a general solution.

Key Concepts

Logistic Differential Equation

The logistic differential equation models population growth with a carrying capacity, expressed as dP/dt = rP(1 - P/K). Here, P(t) is the population at time t, r is the growth rate, and K is the maximum sustainable population. Understanding this equation is essential to analyze how populations grow and stabilize over time.

Verification by Substitution

Verification by substitution involves plugging a proposed solution into the original differential equation to check if it satisfies the equation. This method confirms the correctness of the solution without solving the equation from scratch, ensuring that the given function is indeed a solution.

General Solution and Arbitrary Constants

The general solution of a differential equation includes arbitrary constants representing a family of solutions. In this case, the constant C adjusts the initial condition of the population. Recognizing the role of arbitrary constants helps in understanding how different initial populations affect the solution.