Starting with an initial value of P(0)=55, the population of...

Starting with an initial value of $P(0)=55,$ the population of a prairie dog community grows at a rate of $P^{\prime}(t)=20-t / 5$ (prairie dogs/month), for $0 \leq t \leq 200,$ where $t$ is measured in months. a. What is the population 6 months later? b. Find the population $P(t),$ for $0 \leq t \leq 200$

Starting with an initial value of $P(0)=55,$ the population of
a prairie dog community grows at a rate of $P^{\prime}(t)=20-t / 5$ (prairie dogs/month), for $0 \leq t \leq 200,$ where $t$ is measured in months.
a. What is the population 6 months later?
b. Find the population $P(t),$ for $0 \leq t \leq 200$

Key Concepts

Initial Value Problem

An initial value problem involves a differential equation along with an initial condition that specifies the value of the unknown function at a particular point. This information is essential because it allows for the determination of a unique solution to the differential equation by helping to find the constant of integration after performing the integration process.

Differential Equation

A differential equation is an equation that involves the derivative of an unknown function. It represents how the function changes over time or another variable. In this context, the population of prairie dogs is described by a rate function, making it necessary to understand and solve the differential equation to determine the actual population at any given time.

Integration

Integration is the process of finding the antiderivative of a function, which essentially reverses differentiation. When solving a differential equation, integration is used to determine the original function from its rate of change. The antiderivative includes an arbitrary constant that is later determined using the given initial condition.

Constant of Integration

The constant of integration arises when an indefinite integral is computed, representing the family of all antiderivatives. In an initial value problem, this constant is determined by substituting the initial condition into the general solution, resulting in a specific solution that fits the problem's context.

Population Growth Model

A population growth model is a mathematical representation of how a population changes over time, often expressed using differential equations. These models can incorporate various factors affecting growth, such as birth and death rates or external influences. In the given problem, the linear rate function describes the changing growth rate over time, making it a practical example of applying a population growth model to a real-world scenario.

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