MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §9.4 ODE Analytics

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.3 Differential Equation Applications  Any QUESTIONS About HomeWork §9.3 → HW

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 3 Bruce Mayer, PE Chabot College Mathematics §9.4 Learning Goals  Analyze solutions of differential equations using slope fields  Use Euler’s method for approximating solutions of initial value problems

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 5 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 6 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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:

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 9 Bruce Mayer, PE Chabot College Mathematics Slope Fields  Of course this Variable Separable ODE can be easily solved analytically

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 10 Bruce Mayer, PE Chabot College Mathematics Slope Fields  Example  For the given slope field, sketch two approximate solutions – one of which is passes through(4,2): Solve ODE Analytically using using (4,2) BC Soln

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 11 Bruce Mayer, PE Chabot College Mathematics Slope Field Identification C In order to determine a slope field from a differential equation, we should consider the following: isoclines i) If isoclines (points with the same slope) are along horizontal lines, then DE depends only on y ii) Do you know a slope at a particular point? iii) If we have the same slope along vertical lines, then DE depends only on x iv) Is the slope field sinusoidal? v) What x and y values make the slope 0, 1, or undefined? vi) dy/dx = a( x ± y ) has similar slopes along a diagonal. vii) Can you solve the separable DE? 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ 7. _____ 8. _____ Match the correct DE with its graph: AB C E G D F H H B F D G E A

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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c

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Demand Slope Field  SOLUTION:  Calculate some Slope Values from

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 14 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 15 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Demand Slope Field  Removing ABS Bars  Or  Now use Boundary Value ($2k/unit,0.5)

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Demand Slope Field  Graph for  This is VERY SIMILAR to the Slope Field Graph Sketched Before

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 18 Bruce Mayer, PE Chabot College Mathematics 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 y 1 as

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 19 Bruce Mayer, PE Chabot College Mathematics Numerical Solution - 1  Notation  Exact Numerical Method (impossible to achieve) by Forward Steps  Now Consider y n+1 tntn ynyn t n+1 t tt

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 20 Bruce Mayer, PE Chabot College Mathematics Numerical Solution - 2  The diagram at Left shows that the relationship between y n, y n+1 and the CHORD slope y n+1 tntn ynyn t n+1 t tt  The problem with this formula is we canNOT calculate the CHORD slope exactly We Know Only Δt & y n, but NOT the NEXT Step y n+1 The Analyst Chooses Δt Chord Slope Tangent Slope

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 21 Bruce Mayer, PE Chabot College Mathematics Numerical Solution -3  However, we can calculate the TANGENT slope at any point FROM the differential equation itself  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

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 22 Bruce Mayer, PE Chabot College Mathematics Euler Method – 1 st Order ODE  Solve 1 st Order ODE with I.C.  ReArranging  Use: [Chord Slope]  [Tangent Slope at start of time step]  Then Start the “Forward March” with Initial Conditions

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Euler Estimate  Consider 1 st Order ODE with I.C.  Use The Euler Forward-Step Reln  See Next Slide for the 1 st Nine Steps For Δt = 0.1  But from ODE  So In This Example:

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 24 Bruce Mayer, PE Chabot College Mathematics Euler Exmple Calc ntntn ynyn f n = – y n +1 y n+1 = y n +  t f n Plot Slope

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 25 Bruce Mayer, PE Chabot College Mathematics Euler vs Analytical  The Analytical Solution

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 26 Bruce Mayer, PE Chabot College Mathematics Analytical Soln  Let u = −y+1  Then  Sub for y & dy in ODE  Separate Variables  Integrate Both Sides  Recognize LHS as Natural Log  Raise “e” to the power of both sides

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 27 Bruce Mayer, PE Chabot College Mathematics Analytical Soln  And  Thus Soln u(t)  Sub u = 1−y  Now use IC  The Analytical Soln

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 28 Bruce Mayer, PE Chabot College Mathematics ODE Example:  Euler Solution with ∆t = 0.25, y(t=0) = 37  The Solution Table

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 29 Bruce Mayer, PE Chabot College Mathematics Compare Euler vs. ODE45 Euler SolutionODE45 Solution Euler is Much LESS accurate

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 30 Bruce Mayer, PE Chabot College Mathematics Compare Again with ∆t = Euler SolutionODE45 Solution Smaller ∆T greatly improves Result

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 31 Bruce Mayer, PE Chabot College Mathematics 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)')

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 32 Bruce Mayer, PE Chabot College Mathematics MatLAB Command Window for ODE45 >> dydtfcn 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')

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 33 Bruce Mayer, PE Chabot College Mathematics 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.

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 34 Bruce Mayer, PE Chabot College Mathematics Example  Euler Approximation  Use I.C. to calculate the Initial Slope  Use this slope to Project to the NEW value of y n+1 = y n + Δ y :  Then the NEW value for y:

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 35 Bruce Mayer, PE Chabot College Mathematics Example  Euler Approximation  Tabulating the remaining Calculations  The table then DEFINES y = f ( t )  Thus, for example, y ( t =0.3) = 1.685

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 36 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.4 P32 Population Extinction

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 37 Bruce Mayer, PE Chabot College Mathematics All Done for Today Carl Runge Carl David Tolmé Runge Born: 1856 in Bremen, Germany Died: 1927 in Göttingen, Germany

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 38 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 39 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 40 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-14_sec_9-4_ODE_SlopeFields_Euler.pptx 41 Bruce Mayer, PE Chabot College Mathematics