Wednesday, February 28, 2007

9.1 Modeling with Differential Equations

9.1 - Modeling with Differential Equations



REVIEW
Differential equations discuss rates of change, like a population that changes according to what the limits on the population are or what the population already is.





First order: Here, we have a derivative that is equal to the original function times a constant. Thus, a likely equation for P would be Second order: y’’+y’=x+y
As the number of derivatives increases (and by that I mean first derivative to a second to a third...) , the order does, too. So, because there are both a first derivative and a second derivative in the above equation, it is a second order derivative. To solve it, you must not only find the defintion of the first derivative in terms of x for substituion, but also that of the second derviative (I do not think we will have to be able to do this, though).


Logistic growth:

As you might remember from Precalculus, many populations in the real world begin increasing rapidly, and then gradually level off until it appears not to be increasing at all. That is the curve of a logistic function. One can see that if the population P exceeds the carrying capacity C, then what is in the parentheses will be negative, so dP/dt will be less than zero, and the population will decrease. What happens when we see the graph leveling off is that P is approaching C, so what is inside the parentheses will almost equal zero, thus the dP/dt is almost zero, meaning the population is hardly changing.




EXAMPLES
There are two types of problems…

First, we will be provided with a potential solution to test.



This will be tested by finding y’ and plugging it into the equation to, hopefully, find that there are infinite solutions.
Then, we will have initial value problems (IVP) which is a bit like implicit differentiation in reverse. The problem provides us with an equation for which we find the general solution with a C-value, and another equation so we have the capability for solving for y – solve the differential equation for the general solution, then use an initial value to solve for the specific solution.



Provided: 1. Separate the values, isolating x's and y's on different sides of the equal sign 2. Find the antiderivatives of both sides. (You only have to use one C) And, now you have the general equation.3. Solve for C.4. Write the specific equation.

LINKS
Population dynamics and practice

Paul’s (really detailed) math notes


REMINDER
Laurie, you bum, you are next. Unless sum (hah :( ) elaborate switching process has thrown everything off.


PERSONALIZATION
Blind Date - a math love story

CILLIAN!!!!

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