Numerical Solution of y'=F(x)Example: y'=2x+1, y(0)=1 Symbolic solution y=x^2 + x + 1. Dot table. Connect the dots graphic. How to draw a graphic without knowing the solution equation for y. Making the dot table by approximation of the integral of F(x). Rectangular rule. Dot table steps for h=0.1. Answers: (x,y) = [0, 1], [.1, 1.1], [.2, 1.22], [.3, 1.36], [.4, 1.52], [.5, 1.70], [.6, 1.90], [.7, 2.12], [.8, 2.36], [.9, 2.62], [1.0, 2.90] The exact answers for y(x)=x^2+x+1 are (x,y) = [0., 1.], [.1, 1.11], [.2, 1.24], [.3, 1.39], [.4, 1.56], [.5, 1.75], [.6, 1.96], [.7, 2.19], [.8, 2.44], [.9, 2.71], [1.0, 3.00]

Maple support for making a connect-the-dots graphic.Example: L:=[[0., 1.], [2,3], [3,-1], [4,4]]; plot(L);: connect-the-dots graphic (11.2 K, jpg, 12 Sep 2010)JPG Image

Maple code for the RECT rule, applied to quadrature problem y'=2x+1, y(0)=1.# Quadrature Problem y'=F(x), y(x0)=y0. # Group 1, initialize. F:=x->2*x+1: x0:=0:y0:=1:h:=0.1:Dots:=[x0,y0]:n:=10: # Group 2, repeat n times. RECT rule. for i from 1 to n do Y:=y0+h*F(x0); x0:=x0+h:y0:=Y:Dots:=Dots,[x0,y0]; od: # Group 3, display dots and plot. Dots; plot([Dots]);

Rect, Trap, Simp rules from calculusIntroduction to the Euler, Heun, RK4 rules from this course.Example:y'=3x^2-1, y(0)=2 with solution y=x^3-x+2.Example:y'=2x+1, y(0)=3 with solution y=x^2+x+3. Dot tables, connect the dots graphic. How to draw a graphic without knowing the solution equation for y. Key example y'=sqrt(x)exp(x^2), y(0)=2. Challenge: Can you integrate sqrt(x) exp(x^2)? Making the dot table by approximation of the integral of F(x). Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.

Example for your study:Problem: y'=x+1, y(0)=1 It has a dot table with x=0, 0.25, 0.5, 0.75, 1 and y= 1, 1.25, 1.5625, 1.9375, 2.375. The exact solution y = 0.5(1+(x+1)^2) has values y=1, 1.28125, 1.625, 2.03125, 2.5000. Determine how the dot table was constructed and identify which rule, either Rect, Trap, or Simp, was applied.

Euler, Heun, RK4 algorithmsComputer implementation in maple Geometric and algebraic ideas in the derivations. Numerical Integration Numerical Solutions of DE RECT Euler TRAP Heun [modified Euler] SIMP Runge-Kutta 4 [RK4] Numerical work maple L3.1, L3.2, L3.3, L4.1, L4.2, L4.3 will be submitted after Semester Break. No numerical problems from ch 2 are assigned. All discussion of maple programs will be based in the TA session [Laura and Michal]. There will be one additional presentation of maple lab details in the main lecture. The examples used in maple labs 3, 4 are the same as those in exam review problems ER-1, ER-2. Each has form dy/dx=f(x,y) and requires a non-quadrature algorithm, e.g., Euler, Heun, RK4.

Numerical Solution of y'=f(x,y)Two problems will be studied, in maple labs 3, 4. First problem y' = -2xy, y(0)=2 Symbolic solution y = 2 exp(-x^2) Second problem y' = (1/2)(y-1)^2, y(0)=2 Symbolic solution y = (x-4)/(x-2) The work begins in exam review problems ER-1, ER-2, both due before the first midterm exam. The maple numerical work is due much later, after Semester Break.: Problems ER-1, ER-2 (109.2 K, pdf, 17 Aug 2010)Exam Review

ExamplesWeb references contain two kinds of examples. The first three are quadrature problems dy/dx=F(x). The fourth is of the form dy/dx=f(x,y), which requires a non-quadrature algorithm like Euler, Heun, RK4. y'=3x^2-1, y(0)=2, solution y=x^3-x+2 y'=exp(x^2), y(0)=2, solution y=2+int(exp(t^2),t=0..x). y'=2x+1, y(0)=3 with solution y=x^2+x+3. y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).

- Introduction: Maple Labs 3 and 4, due after Semester Break.

Maple lab 3 F2010. Numerical DE (152.5 K, pdf, 17 Aug 2010)

Maple lab 4 F2010. Numerical DE (138.2 K, pdf, 17 Aug 2010)

- References for numerical methods:

How to use maple at home (4.0 K, txt, 06 Jan 2010)

Maple lab 3 symbolic solution, ER-1 solution. (184.6 K, jpg, 08 Feb 2008)

F2010 notes on numerical DE report for Ch2 Ex 10 (97.9 K, pdf, 17 Aug 2010)

F2010 notes on numerical DE report for Ch2 Ex 12 (108.4 K, pdf, 17 Aug 2010)

F2010 notes on numerical DE report for Ch2 Ex 4 (97.8 K, pdf, 17 Aug 2010)

F2010 notes on numerical DE report for Ch2 Ex 6 (125.7 K, pdf, 17 Aug 2010)

Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)

The work for book sections 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4. The numerical work using Euler, Heun, RK4 appears in L3.1, L3.2, L3.3. The actual symbolic solution derivation and answer check were submitted as Exam Review ER-1. Confused? Follow the details in the next link, which duplicates what was done in ER-1.

Sample symbolic solution report for 2.4-3 (22.6 K, pdf, 19 Sep 2006)

Maple lab 3 reportConfused about what to put in your L3.1 report? Do the same as what appears in the sample report for 2.4-3.

Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)

Include the hand answer check. Include the maple code appendix. Then fill in the table in maple Lab 3, by hand. The example shows a hand answer check and the maple code appendix.Download all .mws maple worksheets to disk, then run the worksheet in xmaple.In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...". Some browsers require SHIFT and then mouse-click. Open the saved file in xmaple or maple. Extension .mws [or .mpl] allows interchange between different versions of maple. Mouse copies of the worksheet pasted into email allow easy transfer of code between versions of maple.

Linear Algebraic Equations sections 3.1, 3.2Frame sequences Toolkit: combo, swap, multiply Plane and space geometry The three possibilities Unique solution No solution Infinitely many solutions Method of elimination Example for a unique solution x + 2y = 1 x - y = -2 Example for no solution x + 2y = 1 x + 2y = 2 Example for infinitely many solutions x + 2y = 1 0 = 0 Parameters in the general solution Differential equations example, problem 3.1-26 y'' -121y = 0, y(0)=44, y'(0)=22 General solution given: y=A exp(11 x) + B exp(-11 x) Substitute y into y(0)=44, y'(0)=22 to obtain a 2x2 system for unknowns A,B that has the unique solution A=23, B=21.

Maple lab 2 problem 1Discussion: Option 1: Freezing pipes maple lab 2 Problem: u' + ku = kA(t) Integration methods Tables Maple Answer check by computer

- Links for maple lab 2:

For more on superposition y=y_p + y_h, see Theorem 2 in the link

Linear DE part I (152.7 K, pdf, 07 Aug 2009)

For more about home heating models, read the following links.

Lecture: 3.1, 3.2, 3.3Frame sequences Toolkit: combo, swap, multiply Plane and space geometry The three possibilities Unique solution No solution Infinitely many solutions Lead variable Free variable Signal equation Echelon form The last frame test The last frame algorithm A detailed account of the three possibilities Unique solution == zero free variables No solution == signal equation Infinitely many solutions == one+ free variables

How to solve a linear system using the toolkitToolkit: swap, combo, mult Toolkit operations neither create nor destroy solutions! Frame sequence examples Computer algebra systems and error-free frame sequences. How to program maple to make a frame sequence without errors.

Solved ProblemsExample 4 in 3.2Back-substitution should be presented as combo operations in a frame sequence, not as isolated, incomplete algebraic jibberish. Technically, back-substitution is identical to applying the frame sequence method to variables in reverse order. The textbook observes that an echelon matrix as frame one is a special case, when only combo operations are required to determine the last frame. Then, and only then, does the last frame algorithm apply to write out the general solution.Problem 3.2-24The book's answer is wrong, it should involve k-4. See references on 3 possibilities with symbol k.

Murphy's Lecture, time permitting: Maple Lab 2, problem 1,2,3 details.

Sample Exam: Exam 1 key from S2010. See also F2009, exam 1.

Exams and exam keys for the last 5 years (17.7 K, html, 18 Dec 2010)

Maple lab 2 problem 1Discussion: Option 1: Freezing pipes maple lab 2 Problem: u' + ku = kA(t) Integration methods Tables Maple Answer check by computer

- Links for maple lab 2:

For more on superposition y=y_p + y_h, see Theorem 2 in the link

Linear DE part I (152.7 K, pdf, 07 Aug 2009)

For more about home heating models, read the following links.

ReviewLast frame test. The RREF of a matrix. Last frame algorithm. Scalar form of the solution. Translation of equation models Scalar equations to augmented matrix Augmented matrix to scalar equations Matrix toolkit: Combo, swap and multiply