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

1. Chabot Mathematics §9.1 ODEModels Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 8.3 Review § • Any QUESTIONS About • §8.3 → TrigonoMetric Applications • Any QUESTIONS About HomeWork • §8.3 → HW-12 “TriAnguLation

3. §9.1 Learning Goals • Solve “variable separable” differential equations and initial value problems • Construct and use mathematical models involving differential equations • Explore learning and population models, including exponential and logistic growth

4. ReCall Mathematical Modeling • DEVELOP MATH EQUATIONS that represent some RealWorld Process • Almost always involves some simplifying ASSUMPTIONS • SOLVE the Math Equations for the quanty/quantities of Interest • INTERPRET the Solution – Does it MATCH the RealWorld Results?

5. Differential Equations • A DIFFERENTIAL EQUATION is ANY equation that includes at least ONE calculus-type derivative • ReCall that Derivatives are themselves the ratio “differentials” such as dy/dx or dy/dt • TWO Types of Differential Equations • ORDINARY (ODE) → Exactly ONE-Each INdependent & Dependent Variable • PARTIAL (PDE) → Multiple Independent Variables

6. Differential Equation • ODE Examples • ODEs Covered in MTH16 • PDE’s • PDEs NOT covered in MTH16

7. Terms of the (ODE) Trade • a SOLUTION to an ODE is a FUNCTION that makes BOTH SIDES of the Original ODE TRUE at same time • A GENERAL Solution is a Characterization of a Family of Solutions • Sometimes called the Complementary Solution

8. Terms of the (ODE) Trade • ODEs coupled with side conditions are called • Initial Value Problems (IVP) for a temporal (time-based) independent variable • Boundary Value Problems (BVP) for a spatial (distance-based) independent variable • a Solution that the satisfies the complementary eqn and side-condition is called the Particular Solution

9. Example  Develop Model • After being implanted in a mouse, the growth rate in volume of a human colon cell over time is proportional to the difference between a maximum size M and the cell’s current volume V • Write a differential equation in terms of V, M, t, and/or a constant of proportionality that expresses this rate of change mathematically.

10. Example  Develop Model • SOLUTION: • Translate the Problem Statement Phrase-by-Phrase “…the growth rate in volume of a human colon cell over time is proportional to the difference between a maximum size M and the cell’s current volume V…” • Build the ODE Math Model

11. Separation of Variables • The form of a “Variable Separable” Ordinary Differential Equation • Find The General Solution by SEPARATING THE VARIABLES and Integrating

12. Example  Solve Mouse ODE • Consider the differential equation for cell growth constructed previously. • The colon cell’s maximum volume is 14 cubic millimeters • The cell’sits current volume is 0.5 cubic millimeters • Six days later the cell has volume increases 4 cubic millimeters. • Find the Particular Solution matching the above criteria.

13. Example  Solve Mouse ODE • SOLUTION: • ReCall the ODE Math Model • From the Problem Statement, the Maximum Volume  • Using M = 14 in the ODE State the Initial Value Problem as WithTimeBasedValues

14. Example  Solve Mouse ODE • The ODE is separable, so isolate factors that can be integrated with respect to V and those that can be integrated with respect to t • Then the Variable-Separated Equation

15. Example  Solve Mouse ODE • IntegrateBoth Sidesans Solve

16. Example  Solve Mouse ODE • At this Point have 2 Unknowns: • Use the Given Time-Points (initial values) to Generate Two Equations in Two Unknowns • Using V(0) = 0.5 mm3

17. Example  Solve Mouse ODE • Now use the other Time Point: • Thus the particular solution for the volume of the cell after t days is

18. Example  Verify ODE Solution • Verify ODE↔Solution Pair • ODE • Solution • Take Derivative of Proposed Solution

19. Example  Verify ODE Solution • Sub into ODE the dB/dt relation • Which by Transitive Property Suggests • Thus by Calculus and Algebra on the ODE • Which IS the ProPosed Solution for B 

20. WhiteBoard Work • Problems From §9.1 • P52 → Work Efficiency

21. All Done for Today GolfBallFLOWSeparation

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