MATH241 Lecture Notes - Lecture 34: Relative Growth Rate, Logistic Function
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MATH241 - Lecture 34 - Separation of Variables and Models for Population Growth
Chapter 9: Separation of Variables and Models for Population Growth
Differential Equation: An equation that contains an unknown function and one or more its
derivatives
The order of a differential equation is the order of the highest derivative that occurs in the
equation
Separable Equation: A first order differential equation in which the expression for
can be
factored as a function of times a function of
In other words, 𝑑𝑑
or 𝑑𝑑
Such an expression can be “separated” into a function of and a function of . Thus it can be
rewritten as
Such an equation can be integrated in the following manner:
Recall the model of population growth 𝑑𝑑
This is separable, so it can be rewritten as 𝑑𝑑
→
→
→
Thus the solution of the initial value problem (IVP) 𝑑𝑑
, 0 0 is 0
The Logistic Model
Sometimes we also want to reflect the fact that relative growth rate decreases as increases
and becomes negative if ever exceeds its carrying capacity
Such a model is the logistic differential equation 𝑑𝑑
1
The logistic equation is separable