Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §9.3 ODEApplications Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 9.2 Review § • Any QUESTIONS About • §9.2 First Order, Linear, Ordinary Differential Equations • Any QUESTIONS About HomeWork • §9.2 → HW-14

3. §9.3 Learning Goals • Use differential equations to model applications involving public health, orthogonal trajectories, and finance. • Explore the predator-prey model

4. Example  Model Epidemic • Consider a population of individuals amidst an outbreak of some disease, with fractions of the total population Ssusceptible, Iimmune, and Ddiseased. • One model for the spread of an epidemic is that the rate of change in the fraction of diseased individuals is jointly proportional to the number susceptible and diseased individuals.

5. Example  Model Epidemic • (a) Supposing that the size of the population and value of I are fixed, write a differential equation modeling the change in fraction of diseased individuals over time. • (b) If a constant 10% of the total population is immune, initially 0.04% of the population has the disease, and after one week 0.1% of the population has the disease, find an equation giving the fraction of the population that is diseased after t days.

6. Example  Model Epidemic • SOLUTION: • (a) As with many modeling problems, start by carefully translating the English Words into mathematics: • “the rate of change in the fraction of diseased individuals is jointly proportional to the fraction of susceptible and diseased individuals”

7. Example  Model Epidemic • Since EveryOne is Either Sick, Immune, or Diseases then S, I, and D are Percentages that must add up to 100%: • Then the fraction of susceptible individuals “leftover” after immune and diseased individuals are accounted for. • Thus the revised ODE

8. Example  Model Epidemic • (b) With a constant 10% Immune, solve an initial value problem for the differential equation • Note that this eqn is Variable Separable

9. Example  Model Epidemic • Integrating theSeparated eqn: • The complex AntiDerivative can be accomplished using the Table of Integrals #6 from Section 6.1: • Use Algebra on the above Equation to solve for D(t)

10. Example  Model Epidemic • Working to Isolate D(t) • Letting • Find • Remove ABS bars as expect D<0.9 (90%)

11. Example  Model Epidemic • Working to Isolate D(t) • Now Use Initial Condition: • In the D(t) eqn • Next, Solve the Above Eqn for Constant Exponential PreFactor, A

12. Example  Model Epidemic • Solving for A • Use A in D(t) • Now Use the 2nd Temporal Condition

13. Example  Model Epidemic • By 2nd I.C.

14. Example  Model Epidemic • Use the Values of A & k to construct the completed Function, D(t)

15. Example  Linked ODE’s • The price of gasoline and the number of purchased electric cars depend on one another. Assume that the rate of change in price of gasoline is a decreasing linear function of the price of electric cars. Similarly, the rate of change in the number of electric cars purchased is an increasing linear function of the price of gasoline. • (a) Model the relationships in rates of change as linked differential equations

16. Example  Linked ODE’s • (b) Say that gasoline increases by $1 per year in the absence of electric cars and the rate decreases by 5 cents for each additional thousand cars that are produced. Also, say that if gas were (magically) priced at $0/gal, there would be a growth rate of 3 thousand, with the rate increasing by 4 thousand for each $1 increase in the unit price of gas. Solve the differential equations implicitly.

17. Example  Linked ODE’s • SOLUTION: • (a) Again Very CareFully Translate the Word-Statement to Math Relations • “the rate of change in price of gasoline is a decreasing linear function of the price of electric cars” 

18. Example  Linked ODE’s • Now construct the differential equation for the change in the car price: • “the rate of change in the number of electric cars purchased is an increasing linear function of the price of gasoline” 

19. Example  Linked ODE’s • Thus have constructed TWO ODE’s in for G(t) and C(t) with 4 unknown constants: a, b, c, and f • More translation is in order to find values of the constants in the two ODEs:

20. Example  Linked ODE’s • “gasoline increases by $1 per year in the absence of electric cars and the rate decreases by 5 cents for each additional thousand NonGasoline cars produced” • Also • “if gas were priced at $0/gal, there would be a growth rate of 3 thousand, with the rate increasing by 4 thousand for each $1 increase in the unit price of gas”

21. Example  Linked ODE’s • Instead of Finding G(t) and C(t) determine an Implicit relation between the two dependent variables • Note that G depends on C, and C depends on G; i.e. the Equations are Coupled • Try one of

22. Example  Linked ODE’s • Find dG/dC: • Separating the Variables • Finding the AntiDerivatives

23. Example  Linked ODE’s • The FinalG & C Relation: • This relationship does not define a function, but it nevertheless is a predictable relation between gasoline price and electric car sales • The graph in the CG plane is an ellipse

24. WhiteBoard Work • Problems From §9.3 • P16: Atmospheric Pressure • P28: Predator-Prey

25. All Done for Today PredatorvsPrey Wolves vs. Elk

26. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –