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

1. Chabot Mathematics §9.4 ODEAnalytics Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 9.3 Review § • Any QUESTIONS About • §9.3 Differential Equation Applications • Any QUESTIONS About HomeWork • §9.3 → HW-15

3. §9.4 Learning Goals • Analyze solutions ofdifferential equationsusing slope fields • Use Euler’s methodfor approximating solutions of initialvalue problems

4. Slope Fields • Recall that indefinite integration, or AntiDifferentiation, is the process of reverting a function from its derivative. • In other words, if we have a derivative, the AntiDerivative allows us to regain the function before it was differentiated – EXCEPT for the CONSTANT, of course. • Given the derivative dy/dx= f ‘(x) then solving for y (or f(x)), produces the General Solution of a Differential Eqn

5. Slope Fields • AntiDifferentiation(Separate Variables) Example • Let: • Then Separating the Variables: • Now take the AntiDerivative: • To Produce the General Solution: • This Method Produces an EXACT and SYMBOLIC Solution which is also called an ANALYTICAL Solution

6. Slope Fields • Slope Fields, on the other hand, provide a Graphical Method for ODE Solution • Slope, or Direction, fields basically draw slopes at various CoOrdinates for differing values of C. • Example: The Slope Field for ODE

7. Slope Fields • slope field describes several different parabolas based on varying values of C • Slope Field Example: create the slope field for the Ordinary Differential Eequation:

8. Slope Fields • Note that dy/dx = x/y calculates the slope at any (x,y) CoOrdinate point • At (x,y) = (−2, 2), dy/dx = −2/2 = −1 • At (x,y) = (−2, 1), dy/dx = −2/1 = −2 • At (x,y) = (−2, 0), dy/dx = −2/0 = UnDef. • And SoOn • Produces OutLine of a HYPERBOLA

9. Slope Fields • Of course this Variable Separable ODE can be easily solved analytically

10. Slope Fields  Example • For the given slope field, sketch two approximate solutions – one of which is passes through(4,2): • Solve ODE Analytically usingusing (4,2) BC Soln

11. Slope Field Identification Match the correct DE with its graph: • In order to determine a slope field from a differential equation, we should consider the following: • If isoclines (points with the same slope) are along horizontal lines, then DE depends only on y • Do you know a slope at a particular point? • If we have the same slope along vertical lines, then DE depends only on x • Is the slope field sinusoidal? • What x and y values make the slope 0, 1, or undefined? • dy/dx = a(x ±y) has similar slopes along a diagonal. • Can you solve the separable DE? A B H 1. _____ F 2. _____ C D D 3. _____ C 4. _____ E F A 5. _____ G 6. _____ G H E 7. _____ B 8. _____

12. Example  Demand Slope Field • Imagine that the change in fraction of a production facility’s inventory that is demanded, D, each period is given by • Where p is the unit price in $k • Draw a slope field to approximate a solution assuming a half-stocked (50%) inventory and $2k per item, and then • Verify the Slope-Field solution using Separation of Variables. • c

13. Example  Demand Slope Field • SOLUTION: • Calculate some Slope Values from

14. Example  Demand Slope Field • An approximate solution passing through (2,0.5) with slope field on the window 0 < x < 3 and 0 < y < 1

15. Example  Demand Slope Field • Find an exact solution to this differential equation using separation of variables: • Remove absolute-value and then change signs as inventory demanded satisfies: 0≤ D ≤1

16. Example  Demand Slope Field • Removing ABS Bars • Or • Now use Boundary Value ($2k/unit,0.5)

17. Example  Demand Slope Field • Graph for • This is VERY SIMILAR to the Slope Field Graph Sketched Before

18. Numerical ODE Solutions • Next We’ll “look under the hood” of NUMERICAL Solutions to ODE’s • The BASIC Game-Plan for even the most Sophisticated Solvers: • Given a STARTING POINT, y(0) • Use ODE to find dy/dt at t=0 • ESTIMATE y1 as

19. Notation Numerical Solution - 1 • Exact Numerical Method (impossible to achieve) by Forward Steps yn+1 yn • Now Consider t tn tn+1 Dt

20. Numerical Solution - 2 • The diagram at Left shows that the relationship between yn, yn+1 and the CHORD slope yn+1 Tangent Slope yn Chord Slope • The problem with this formula is we canNOT calculate the CHORD slope exactly • We Know Only Δt & yn, but NOT the NEXT Step yn+1 t tn tn+1 Dt The AnalystChooses Δt

21. However, we can calculate the TANGENT slope at any point FROM the differential equation itself Numerical Solution -3 • The Basic Concept for all numerical methods for solving ODE’s is to use the TANGENT slope, available from the R.H.S. of the ODE, to approximate the chord slope • Recognize dy/dt as the Tangent Slope

22. Solve 1st Order ODE with I.C. Euler Method – 1st Order ODE • ReArranging • Use: [Chord Slope]  [Tangent Slope at start of time step] • Then Start the “Forward March” with Initial Conditions

23. Consider 1st Order ODE with I.C. Example  Euler Estimate • But from ODE • So In This Example: • Use The Euler Forward-Step Reln • See Next Slide for the 1st Nine Steps For Δt = 0.1

24. Euler ExmpleCalc Slope Plot

25. Euler vs Analytical • The Analytical Solution

26. Let u = −y+1 Then Analytical Soln • Integrate Both Sides • Recognize LHS as Natural Log • Sub for y & dy in ODE • Raise “e” to the power of both sides • Separate Variables

27. And Analytical Soln • Now use IC • The Analytical Soln • Thus Soln u(t) • Sub u = 1−y

28. ODE Example: • Euler Solution with ∆t = 0.25, y(t=0) = 37 • The Solution Table

29. Compare Euler vs. ODE45 Euler Solution Euler is Much LESS accurate ODE45 Solution

30. Compare Again with ∆t = 0.025 Euler Solution Smaller ∆T greatly improves Result ODE45 Solution

31. MatLAB Code for Euler % Bruce Mayer, PE % ENGR25 * 04Jan11 % file = Euler_ODE_Numerical_Example_1201.m % y0= 37; delt = 0.25; t= [0:delt:10]; n = length(t); yp(1) = y0; % vector/array indices MUST start at 1 tp(1) = 0; for k = 1:(n-1) % fence-post adjustment to start at 0 dydt = 3.9*cos(4.2*yp(k))^2-log(5.1*tp(k)+6); dydtp(k) = dydt% keep track of tangent slope tp(k+1) = tp(k) + delt; dely = delt*dydt delyp(k) = dely yp(k+1) = yp(k) + dely; end plot(tp,yp, 'LineWidth', 3), grid, xlabel('t'),ylabel('y(t) by Euler'),... title('Euler Solution to dy/dt = 3.9cos(4.2y)-ln(5.1t+6)')

32. MatLAB Command Window forODE45 >> dydtfcn = @(tf,yf) 3.9*(cos(4.2*yf))^2-log(5.1*tf+6); >> [T,Y] = ode45(dydtfcn,[0 10],[37]); >> plot(T,Y, 'LineWidth', 3), grid, xlabel('T by ODE45'), ylabel('Y by ODE45')

33. Example  Euler Approximation • Use four steps of Δt = 0.1 with Euler’s Method to approximate the solution to • With I.C. • SOLUTION: • Make a table of values, keeping track of the current values of t and y, the derivative at that point, and the projected next value.

34. Example  Euler Approximation • Use I.C. to calculate the Initial Slope • Use this slope to Project to the NEW value of yn+1 =yn + Δy: • Then the NEW value for y:

35. Example  Euler Approximation • Tabulating the remaining Calculations • The table then DEFINES y = f(t) • Thus, for example, y(t=0.3) = 1.685

36. WhiteBoard Work • Problems From §9.4 • P32 Population Extinction

37. All Done for Today CarlRunge Carl David Tolmé Runge Born: 1856 in Bremen, Germany Died: 1927 in Göttingen, Germany

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