1. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.2 1st Order ODEs

2. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.1 → Variable Separable Ordinary Differential Equations  Any QUESTIONS About HomeWork §9.1 → HW-13 9.1

3. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 3 Bruce Mayer, PE Chabot College Mathematics §9.2 Learning Goals  Solve first-order linear differential equations and Initial Value Problems (IVP) Boundary Value Problems (BVP)  Explore compartmental analysis with applications to finance, drug administration, and dilution models.

4. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 4 Bruce Mayer, PE Chabot College Mathematics FirstOrder, Linear ODE  The General form of a First Order, Linear Ordinary Differential Equation  Solve the General Equation with Integrating Factor Let Then the ODE Solution

5. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 5 Bruce Mayer, PE Chabot College Mathematics Quick Example  Find Solution to ODE:  The Integrating Factor →  Thus the Solution

6. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Find the Particular Solution for ODE:  Subject to Initial Value:  SOLUTION:  Note that this Eqn is NOT Variable Separable, so ReWrite in General Form

7. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Then the Integrating Factor:  Now Let  Then  Using u and du in integrating Factor  Now t 2 +1 is always positive so:

8. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Using this Integrating Factor find:  Using u and du from before

9. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Then the General Solution by Back SubStitution  ReCall the Initial Condition (IC)  Using IC in Solution

10. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Solve  Finally the Full General Solution

11. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 11 Bruce Mayer, PE Chabot College Mathematics I(x) → How Does it Work?  Multiplication of Both Sides of the ODE by I(x) changes ODE appearance  For Solution This must be of the form

12. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 12 Bruce Mayer, PE Chabot College Mathematics I(x) → How Does it Work?  So that by the PRODUCT Rule  ReCall the I(x) multiplied ODE L.H.S.  Thus by Correspondence need

13. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 13 Bruce Mayer, PE Chabot College Mathematics I(x) → How Does it Work?  Then by Substitution  Then the I(x) multiplied ODE  Which is VARIABLE SEPARABLE

14. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 14 Bruce Mayer, PE Chabot College Mathematics I(x) → How Does it Work?  Or  Then Let:  Using u in the Variable Separated ODE  BackSubbing for u  Let −C 1 = +C

15. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 15 Bruce Mayer, PE Chabot College Mathematics No Need for Memorization  Do Need to Memorize  Only need to find a good I(x) to multiply the ODE so that by the PRODUCT Rule the L.H.S.:  Then can Separate the Variables and Integrate

16. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 16 Bruce Mayer, PE Chabot College Mathematics Key to Integrating Factor  Need  Then Assumes, withOUT loss of generality, that the Constant of Integration is Zero  So Finally the Integrating-Factor Formula

17. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 17 Bruce Mayer, PE Chabot College Mathematics Key to Integrating Factor  For Solution Need:  Next Integrate this ODE  Then Assumes, withOUT loss of generality, that the Constant of Integration is Zero  So Finally the Integrating-Factor Formula

18. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  A 60-gallon barrel containing 20 gallons of simple syrup at 1:1 sugar-to-water ratio is being stirred and filled with pure sugar at a rate of 1 gallon per minute. Unfortunately, a crack in the bottom of the barrel is leaking solution at a rate of 4 oz per minute.  After how long will there be 40 gallons of Pure Sugar in the barrel?

19. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  SOLUTION:  First to set up an equation to model the quantity of sugar in the barrel over time,  Next solve this eqn and find the time at which the desired quantity occurs.  A general Mass Balance for a “Control Volume” Storage Rate = InFlow − OutFlow Storage InFlowOutFlow

20. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  The Pure Sugar Mass Balance Statement  The Model above accounts for modeling the change in pure-sugar quantity, the inflow is 1 Gallon per Minute (1 gpm) or 128 oz per minute.

21. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  The outflow is of the mixed solution, so it leaks at a rate of 4 oz/min, with total quantity of sugar Q(t) and total quantity of solution equal to:  So the concentration of OutFlowing Syrup:

22. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  Now we express the differential equation for the rate of change in sugar quantity:  This ODE is first-order and linear, so it can solved using the general strategy.

23. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  Calculate the Integrating Factor for the ODE  Then the form of the solution

24. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  Use the IC to find the Constant Value Initially there is a 1:1 ratio of water to sugar, so exactly half of the 20 gallons, or 10 gallons (1280 oz), is sugar. Use this Data-Point to find the value of C:

25. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  Finally, find the time at which there are 40 gallons of sugar in the barrel, which happens when y = 40*128 = 5120 oz.

26. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Dilution over Time  This a transcendental (NonAlgebraic) eqn for which there is NO exact solution  Solve using the MuPAD Computer Algebra System (CAS):  In other words, after about 32.6 minutes of pouring and mixing, there will be 20 gallons of pure sugar in the barrel.

27. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 27 Bruce Mayer, PE Chabot College Mathematics MuPAD Calculation  tsoln := 124*t - 5100 + 1327.14/(5+31*t)^(1/31)  numeric::solve(tsoln)

28. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 28 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.2 P51 Glacier Ice Removal Rate

29. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 29 Bruce Mayer, PE Chabot College Mathematics All Done for Today Linear 1 st Order ODEs

30. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 30 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

31. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 31 Bruce Mayer, PE Chabot College Mathematics

32. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 32 Bruce Mayer, PE Chabot College Mathematics

33. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 33 Bruce Mayer, PE Chabot College Mathematics

34. BMayer@ChabotCollege.edu MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 34 Bruce Mayer, PE Chabot College Mathematics