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. Here's the statements for the exam review problems, which review chapter 1 methods to find the symbolic solutions:: Problems ER-1, ER-2 (109.3 K, pdf, 10 Dec 2011)Exam Review

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

Rect, Trap, Simp rules from calculusRECT Replace int(F(x),x=a..b) by rectangle area (b-a)F(a) TRAP Replace int(F(x),x=a..b) by trapezoid area (b-a)(F(a)+F(b))/2 SIMP Replace int(F(x),x=a..b) by quadratic area (b-a)(F(a)+4F(a/2+b/2)+F(b))/6 The Euler, Heun, RK4 rules from this course: how they relate to calculus rules RECT, TRAP, SIMP Numerical Integration Numerical Solutions of DE RECT Euler TRAP Heun [modified Euler] SIMP Runge-Kutta 4 [RK4]Example:y'=3x^2-1, y(0)=2 with solution y=x^3-x+2.Example:y'=2x+1, y(0)=1 with solution y=x^2+x+1. Dot tables, connect the dots graphic. How to draw a graphic without knowing the solution equation for y. What to do when int(F(x),x) has no formula? 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). Accuracy: Rect, Trap, Simp rules have 1,2,4 digits resp.

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]);

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 [Patrick]. 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 (109.3 K, pdf, 10 Dec 2011)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 S2012. Numerical DE (60.0 K, pdf, 09 Dec 2011)

Maple lab 4 S2012. Numerical DE (56.4 K, pdf, 09 Dec 2011)

- References for numerical methods:

How to use maple at home (6.8 K, txt, 28 Dec 2011)

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

S2012 notes on numerical DE report for Ch2 Ex 10 (35.5 K, pdf, 09 Dec 2011)

S2012 notes on numerical DE report for Ch2 Ex 12 (51.8 K, pdf, 09 Dec 2011)

S2012 notes on numerical DE report for Ch2 Ex 4 (35.0 K, pdf, 09 Dec 2011)

S2012 notes on numerical DE report for Ch2 Ex 6 (47.9 K, pdf, 09 Dec 2011)

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

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.

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 (18.9 K, html, 02 May 2012)

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.

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