1. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.1 ODE Models

2. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §8.3 → TrigonoMetric Applications  Any QUESTIONS About HomeWork §8.3 → HW-12 8.3 “TriAnguLation

3. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 4 Bruce Mayer, PE Chabot College Mathematics ReCall Mathematical Modeling 1.DEVELOP MATH EQUATIONS that represent some RealWorld Process Almost always involves some simplifying ASSUMPTIONS 2.SOLVE the Math Equations for the quanty/quantities of Interest 3.INTERPRET the Solution – Does it MATCH the RealWorld Results?

5. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 5 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 6 Bruce Mayer, PE Chabot College Mathematics Differential Equation  ODE Examples ODEs Covered in MTH16  PDE’s PDEs NOT covered in MTH16

7. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 9 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 10 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 11 Bruce Mayer, PE Chabot College Mathematics Separation of Variables  The form of a “Variable Separable” Ordinary Differential Equation  Find The General Solution by SEPARATING THE VARIABLES and Integrating

12. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 13 Bruce Mayer, PE Chabot College Mathematics 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 With TimeBased Values

14. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 14 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Solve Mouse ODE  Integrate Both Sides ans Solve

16. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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 mm 3

17. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 17 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Verify ODE Solution  Verify ODE↔Solution Pair ODE Solution  Take Derivative of Proposed Solution

19. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 19 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 20 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.1 P52 → Work Efficiency

21. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 21 Bruce Mayer, PE Chabot College Mathematics All Done for Today GolfBall FLOW Separation

22. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 22 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

23. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 23 Bruce Mayer, PE Chabot College Mathematics

24. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 24 Bruce Mayer, PE Chabot College Mathematics

25. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 25 Bruce Mayer, PE Chabot College Mathematics

26. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 26 Bruce Mayer, PE Chabot College Mathematics

27. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 27 Bruce Mayer, PE Chabot College Mathematics

28. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 28 Bruce Mayer, PE Chabot College Mathematics

29. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 29 Bruce Mayer, PE Chabot College Mathematics