1. Block 5 Stochastic & Dynamic Systems Lesson 18 – Differential Equations - Methods and Models
Charles Ebeling
University of Dayton

2. Differential Equations
Equations involving derivatives used to solve practical problems in engineering and science.
ENM/MSC applications include:
queuing analysis
reliability
Markov processes
modeling population growth, decline, spread of disease
modeling conflicts

3. Examples of Differential Equations
I get it now. A differential equation has those dy-dx things in it.

4. Ordinary Differential Equations
Single independent variable with no partial derivatives
order – highest ordered derivative
degree – algebraic degree of the highest ordered derivative
solution – a non-derivative relationship between the variables which satisfy the diff. eq. as an identify
Examples:
order 2, degree 1
order 3, degree 2

5. Solutions
General solution – solutions involving n constants for an nth order differential equations as a result of n integrations
Particular solution – obtained from general solution by assigning specific values to the constants
Example – show y = cx2 is a general solution to:

6. Our very next diff. equation
Example – show
is a general solution to:
Who would have thought? It isn’t so bad when you are given the answer.

7. First Order – Linear (1st degree)

8. Solving first-order, linear equations
Variables separable
Homogeneous equations
Exact Equations
Linear in y
Engineering Management
Students enjoying solving a
differential equation problem.

9. Variables Separable - Examples
First he says its separable then he says to integrate. I am totally confused.

10. A Separable Example
from tables

11. An old friend
A = the amount of an initial investment at an interest rate of i compounded continuously

12. Another Example

13. 1st Order Linear Diff Equations – General Solution
An integrating factor? Why it is a factor which when multiplied through a differential equation results in an exact differential equation. But everyone knows that.

14. Example #1

15. Not checking your work is indeed dumb.
Check your work!
Not checking your work is indeed dumb.

16. Example #2

17. Check

18. Putting your math to work…
Dynamic Models
Putting your math to work…

19. Exponential Growth
A population increases at a rate proportional to its size. If a population doubles in one year, how many years before it will be 1000 times its original size?
Let N = population size at time t and N0 = population at t = 0

20. More Population Growth

21. The Logistics Curve
Under exponential growth, a population gets infinitely large as time goes on
In realty, there are limits to growth that will slow down the rate of growth
food supply, predators, overcrowding, disease, etc.
Assume the size of a population, N, is limited to some maximum number M, where
0 < N < M and as NM, dN/dt  0
Desire exponential growth initially but then limits to growth

22. The Model For N small, (M-N) / M is close to 1
and growth is approximately exponential
Then as NM, (M-N)  0 and dN/dt  0

23. Let’s solve for N…
from the old table of integrals:

24. Keep on solving…

25. The Logistics Curve

26. The Maximum Rate of Growth

27. Now find the time at which the maximum growth rate occurs…
substitute M/2 for N
solve for t

28. Time for an example
The new Montgomery County jail can hold a maximum of 800 prisoners. One year ago there were 50 prisoners in the jail and now there are Assuming the jail population follows a logistic function, how many prisoners will there be three years from now?
Let N(t) = the jail population after t years since opening
when t = 0:

29. Still time for an example
I’ll be out of this joint before then.
Max growth rate occurs at t = ln(15)/ln5 =1.7 years

30. A Campus Rumor
In a Midwestern university, having a student population of 10,200, a rumor had been initiated among a class of 30 students that a favorite engineering professor will be winning the Noble prize in economics for his work on multivariable nonlinear profit maximization based upon first and second order partial derivatives. After a week, the rumor had spread around campus to 160 students. When the semester ends 5 weeks later (week 6 since rumor initiation), how many students will know the rumor? Assume the logistics growth process.

31. The rumor is spreading…

32. still spreading…
Yes, it’s all over the campus. He has won the Noble prize in economics. I am so glad I took his course.

33. All good things must come to an end.
Hey, don’t run away from solving these equations
All good things must come to an end.
An engineering professor caught grading student papers

34. ENM 503 students saddened as they approach the end of the course