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

1. Chabot Mathematics §9.2 1stOrder ODEs Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 9.1 Review § • Any QUESTIONS About • §9.1 → Variable Separable Ordinary Differential Equations • Any QUESTIONS About HomeWork • §9.1 → HW-13

3. §9.2 Learning Goals • Solve first-order linear differential equations and • Initial Value Problems (IVP) • Boundary Value Problems (BVP) • Explore compartmentalanalysis with applicationsto finance, drug administration, and dilution models.

4. FirstOrder, Linear ODE • The General form of a First Order, Linear Ordinary Differential Equation • Solve the General Equation with Integrating Factor • Let • Then the ODESolution

5. Quick Example • Find Solution to ODE: • The IntegratingFactor → • Thus theSolution

6. 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. Example  Solve • Then the IntegratingFactor: • Now Let • Then • Using u and du in integrating Factor • Now t2+1 is always positive so:

8. Example  Solve • Using this Integrating Factor find: • Using u and du from before

9. Example  Solve • Then the General Solution by Back SubStitution • ReCall the Initial Condition (IC) • Using IC in Solution

10. Example  Solve • Finally the Full General Solution

11. 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. 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. I(x) → How Does it Work? • Then by Substitution • Then the I(x) multiplied ODE • Which is VARIABLE SEPARABLE

14. I(x) → How Does it Work? • Or • Then Let: • Using u in the Variable Separated ODE • BackSubbing for u • Let −C1 = +C Q.E.D.

15. No Need for Memorization • Do Need toMemorize • 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. 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. 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. 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. 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 InFlow OutFlow

20. 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. 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. 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. Example  Dilution over Time • Calculate the Integrating Factor for the ODE • Then the form of the solution

24. 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. 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. 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. MuPAD Calculation • tsoln := 124*t - 5100 + 1327.14/(5+31*t)^(1/31) • numeric::solve(tsoln)

28. WhiteBoard Work • Problems From §9.2 • P51Glacier IceRemovalRate

29. All Done for Today Linear1st OrderODEs

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