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MA 36600 LECTURE NOTES: WEDNESDAY, JANUARY 14 3 “the rate of change of the size of the population is proportional to the size of the population as well as the distance the population is from its sustainable size.” Another way to say this is dP dt P 1 P K = dP dt = r P 1 P K . This di ff erential equation is called the Logistic Equation . We will see later that its solution is P ( t ) = K P 0 P 0 + ( K P 0 ) e rt . Constructing Mathematical Models. We have seen that a di ff erential equation is an equation involving di ff erentials. There are three key steps to using a di ff erential equation to model a physical situation: #1: Identify the key variables, such as position x , velocity v , population size P , or time t . Decide which variables are independent , such as time
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Unformatted text preview: , such as time t ; and which are dependent , such as position x = x ( t ), velocity v = v ( t ), or population size P = P ( t ). #2: Articulate the principle that underlies the problem under investigation. This can be used to articulate the diFerential equation. ±or example, Newton’s Second Law of Motion states F = m a ; this is a diFerential equation. #3: Identify the initial conditions. ±or example, when dealing with position, we can write x ( t ) = x as the initial position and v ( t ) = v as the initial velocity. Whenever we have (1) a diFerential equation and (2) a list of initial conditions we call this an initial value problem ....
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