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Chap9_Sec1

# Chap9_Sec1 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

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DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9

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Perhaps the most important of all the applications of calculus is to differential equations. DIFFERENTIAL EQUATIONS
When physical or social scientists use calculus, more often than not, it is to analyze a differential equation that has arisen in the process of modeling some phenomenon they are studying. DIFFERENTIAL EQUATIONS

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It is often impossible to find an explicit formula for the solution of a differential equation. Nevertheless, we will see that graphical and numerical approaches provide the needed information. DIFFERENTIAL EQUATIONS
9.1 Modeling with Differential Equations In this section, we will learn: How to represent some mathematical models in the form of differential equations. DIFFERENTIAL EQUATIONS

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In describing the process of modeling in Section 1.2, we talked about formulating a mathematical model of a real-world problem through either: Intuitive reasoning about the phenomenon A physical law based on evidence from experiments MODELING WITH DIFFERENTIAL EQUATIONS
The model often takes the form of a differential equation. This is an equation that contains an unknown function and some of its derivatives. DIFFERENTIAL EQUATION

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This is not surprising. In a real-world problem, we often notice that changes occur, and we want to predict future behavior on the basis of how current values change. MODELING WITH DIFFERENTIAL EQUATIONS
Let’s begin by examining several examples of how differential equations arise when we model physical phenomena. MODELING WITH DIFFERENTIAL EQUATIONS

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One model for the growth of a population is based on the assumption that the population grows at a rate proportional to the size of the population. MODELS OF POPULATION GROWTH
That is a reasonable assumption for a population of bacteria or animals under ideal conditions, such as: Unlimited environment Adequate nutrition Absence of predators Immunity from disease MODELS OF POPULATION GROWTH

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Let’s identify and name the variables in this model: t = time (independent variable) P = the number of individuals in the population (dependent variable) MODELS OF POPULATION GROWTH
The rate of growth of the population is the derivative dP / dt . MODELS OF POPULATION GROWTH

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Hence, our assumption that the rate of growth of the population is proportional to the population size is written as the equation where k is the proportionality constant. dP kP dt = POPULATION GROWTH MODELS Equation 1
Equation 1 is our first model for population growth. It is a differential equation because it contains an unknown function P and its derivative dP / dt. POPULATION GROWTH MODELS

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Having formulated a model, let’s look at its consequences. POPULATION GROWTH MODELS
If we rule out a population of 0, then P ( t ) > 0 for all t So, if k > 0, then Equation 1 shows that: P’ ( t ) > 0 for all t POPULATION GROWTH MODELS

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This means that the population is always increasing.
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