1. Differential Equations A Universal Language
Bethany Caron
Spring 2008
Senior Honors Project

2. Math is not as scary as it seems!
Explain complex mathematical concepts in “non-math” language
Used as a tool in modeling many different fields of study
Pure Mathematics vs. Applied Mathematics

3. Derivatives Measures the instantaneous rate of change of a function
Denoted by f ’ (x) or df/dx

4. Types of Differential Equations
Ordinary Differential Equations
Mathematical equation involving a function and its derivatives
Involve equations of one single variable
Partial Differential Equations
Involve equations of more than one variable and their partial derivatives
Much more difficult to study and solve

5. Uses of Differential Equations
Study the relationship between a changing quantity and its rate of change
Help solve real life problems that cannot be solved directly
Model real life situations to further understand natural and universal processes
Model the behavior of complex systems

6. The Process Real World Situation Formulation Interpretation
Mathematical
Analysis
Mathematical Model
Mathematical Results

7. Practical Applications
Physics
light and sound waves
Newton’s Laws of Motion
radioactive decay
oscillation
Economics and Finance
equity markets
net worth
Biological Sciences
predator / prey
population growth

8. Practical Applications
Engineering
bridge design
electrical circuits
Astronomy
celestial motion
Chemistry
interaction between neurons
Newton’s Law of Cooling
Forensics - time of death
temperature of meat

9. Kermack-McKendrick Model: Modeling Infectious Disease
Famous SIR model
S: Susceptible People
I: Infected People
R: Removed People
Models contagious disease in a specific population over time
Simplifies spread and recovery of disease
Created to model epidemics
Plague
Cholera
Flu
Measles
Tuberculosis

10. Kermack-McKendrick Model
Assumes a fixed and closed population
Model in 2-dimensional space
Need for partial differential equations arises
Much more difficult to understand and solve
More effective and accurate model

11. Newton’s Law of Heating and Cooling
Rate of change of temperature
Proportional to difference between temperature of object T and temperature of environment Ta
Law of Heating
Positively correlated
Law of Cooling
Negatively correlated
Forensics
Time of death
Temperature of body at different times

12. Newton’s Law of Heating and Cooling
Cooling a cup of coffee
Defrosting food
Cooking time for meat

13. Conclusion
Challenges
Explaining complex mathematical ideas
Finding and understanding solutions
“Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.” – David Hilbert