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Unformatted text preview: MATH 310 Differential Equations Homework Exercises, Set 6 Summer 2011 Quiz on Wednesday, 26 October 2011 Course Information is on WebCT Textbook references are to: Boyce and DiPrima "Elementary Differential Equations and Boundary Value Problems" (9th edition). Reading: Sections 2.4, 2.5, 2.6 Problems to prepare for quiz (no need to hand in): Section 2.7, # 21 Euler's method (see also the additional problem); first read through #20. Recall limh0 (1 + h)t/h from calculus. . . # 8, 10, 15, 17, 20, 21, 23, 28 Second order linear DEs, real distinct roots of the characteristic equation # 12, 14, 15, 16 General theory Be sure you understand the ideas behind #1415 clearly (linearity!) For #16, one approach is to think about the values of y and y at t = 0 Section 3.1, Section 3.2, Additional problems 1. For the initial-value problem y = 3 + t - y, y(0) = 1 : (a) Find the analytical solution y(t), and evaluate it at t = 0.1 and t = 0.2. (b) Compute approximate values of the solution at t = 0.1 and t = 0.2 "by hand" (you can use a calculator if needed!) using Euler's Method with step size h = 0.1, writing out the steps in the calculation. Compare with the exact solution from (a) to find the errors. (c) As in (b), compute "by hand" approximations to the solution at t = 0.1 and t = 0.2 using step size h = 0.05, and find the errors. How do the errors compare with those for h = 0.1? [This is a shortened version of Section 2.7 #1.] You are strongly encouraged to look at and work through other problems; the following is a suggested selection of other useful or interesting (optional) exercises: Section 2.7, Section 3.1, # 2, 12, 20 Numerical methods: Euler's method # 1, 4, 11, 12, 16, 22, 27 Second order linear DEs, real distinct roots Do as many of # 118 as needed # 9, 10, 13 General theory of linear equations: existence/uniqueness, linearity Section 3.2, ...
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