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# Week 14 Notes Math 340

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This 4 page Class Notes was uploaded by Susan Ossareh on Sunday April 24, 2016. The Class Notes belongs to Math 340 at Colorado State University taught by in Spring 2016. Since its upload, it has received 7 views. For similar materials see Intro-Ordinary Differen Equatn in Math at Colorado State University.

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Date Created: 04/24/16

th Math 340 Lecture – Introduction to Ordinary Differential Equations – April 25 , 2016 What We Covered: 1. Prepare for Quiz (second to last one guys!) a. This will cover 4.5 and 5.1 2. Course Content – Chapter 5: The Laplace Transform (LT) a. Section 5.1: The Definition of the Laplace Transform i. The whole point of LT is to find the solution to inhomogeneous equations ii. Supposed f(t) is a function of t defined for 0 < ???? < ∞. The Laplace transform of f is the function: ∞ ℒ ???? ???? = ???? ???? = ∫ ????(????)???? −???????????? ???????????? ???? > 0 0 iii. When ???? < ????, the exponent is positive and so the term involving ???? −(????−????)approaches infinity. Thus, the Laplace transform of ???? ???? = ???? ???????? is −(????−????)???? undefined for ???? ≤ ????. When ???? > ????, the term involving ???? converges to zero and therefore: 1 ℒ ????????????)(???? = ???? ???? =) ???? − ???? iv. Also, when a=0 1 ℒ 1 ???? = ???? ???? = ) ???? v. Honestly, the most useful method in solving these problems is integration by parts. You don’t have to memorize them, just understand this definition and how to solve it ∫???????????? = ???????? − ∫???????????? vi. After computing the Laplace transform of f(t)=t you get 1 ℒ ???? ???? = ???? ???? = ) 2 ???? vii. And the same process used earlier can be applied again to compute the Laplace transform of any power t … ???? ????! ℒ ???? )( ???? = ????+1 ???? viii. Piecewise continuous functions 1. A piecewise continuous function Is one that has only finitely many discontinuous and at those points the side limits exist 2. A piecewise differentiable function is a function that has a piecewise continuous derivative ix. Exponential Order 1. A function f(t) is of exponential order if there are constants C and a such that ???????? |???? ???? | ≤ ???????? x. Theorem 1. Suppose f is a piecewise continuous function defined on [0,∞), which is of exponential order. Then the Laplace transform exists for large values of s. Specifically, if |???? ???? | ≤ ???????? , then ℒ(????)(????) exists at least for ???? > ???? b. Section 5.2: Basic Properties of the Laplace Transform i. Proposition (key tool): Suppose y is a piecewise differentiable function of exponential order. Suppose also that y’ is of exponential order. Then for ′ large values of s, ℒ ????( ???? = ????ℒ ???? ???? − ???? 0 = ???????? ???? − ????(0), where Y(s) is the Laplace transform of y ii. Proposition: Suppose that y and y’ are piecewise differentiable and continuous and that y” os piecewise continuous. Suppose that all three are of exponential order. Then ℒ ????( ′′)???? = ???? ℒ ???? ???? − ???????? 0 − ???? 0 = ( ) 2 ???? ???? ???? − ???????? 0 − ????′(0), where Y(s) is the Laplace transform of y. More generally, if y and all of its derivatives up to order k – 1 are piecewise differentiable and continuous, and y (kis piecewise continuous, and all of them has exponential order, then ℒ ( (????))(???? = ???? ℒ ???? ???? − ???? ????−1 ???? 0 −...−???????? (????−2)(0 − ???? (????−1)(0) ???? ????−1 ????−2 ) (????−1) = ???? ???? ???? − ???? ???? 0 −...−???????? (0 − ???? (0) Suggested Homework: Study for quiz Section 5.1: 5, 9, 13, 19 Section 5.2: 22, 25, 26, 32 Math 340 Lab – Introduction to Ordinary Differential Equations – April 26 , 2016 th What We Covered: 1. Course Content – Chapter 5: Laplace Transform a. Section 5.2: Basic Properties of the Laplace Transform i. Basic Table f ℒ(????) 1 1 ???? ???????? ????! ????+1 ???? sin(at) ???? 2 2 ???? + ???? cos(at) ???? ???? + ???? 2 ???????? 1 ???? ???? − ???? ???? (????) ???? ℒ ???? ???? − ???? ????−1???? 0 −...− ???? ????0 (????−1)(0) ii. The LT is linear iii. Proposition: Suppose f and g are piecewise continuous functions of exponential order, and α and β are constants. Then ℒ (???? ???? + ???????? ???? ( ))(???? = ????ℒ ( ????( ) ???? + ????ℒ(???? ???? )(????) ????ℎ???? ???????????????????? ???????? ????ℎ???????? ????ℎ???? ???????????????????????????? ???????????????????????????????????? ???????? ???? ???????????????????????? ???????????????????????????????????????????? ???????? ???????????????????????????????????? ???????????? ???????? ???????????????????????????????? ???????? ???????????????????????? ????ℎ???? ???????????????????????????? ???????????????????????????????????? ???????? ????????????ℎ ???????????????? ???????????????????????????????????????? ???????????? ????ℎ???????? ???????????????????????? ???????? ????ℎ???? ???????????????????????? iv. Proposition: Suppose f is a piecewise continuous function of exponential order. Let F(s) be the Laplace transform of f, and let c be any constant. Then ℒ(???? ???? ????( ) ???? = ????(???? − ????) v. Proposition: Suppose f is a piecewise continuous function of exponential order, and let F(s) be its Laplace transform. Then ( ???????? ????)(???? = −????′(????). More generally, if n is any positive integer, then ℒ(???? ???? ????))(???? = (−1) ???????? (????)(????) b. Section 5.3: Inverse Laplace Transform i. Theorem: Suppose that f and g are continuous functions and that ℒ ???? ???? = ℒ(????)(????) for ???? > ????. Then f(t)=g(t) for all ???? > 0 ii. Definition: If f is a continuous function of exponential order and ℒ ???? ???? = ????(????), then we call f the inverse Laplace transform of F, and −1 write ???? = ℒ (????) iii. Proposition: The inverse Laplace transform is linear. Suppose that ℒ −1(???? = ???? ???????????? ℒ −1(???? = ????. Then for any constants a and b, −1 −1 −1 ℒ (???????? + ???????? = ????ℒ (???? + ????ℒ (???? = ???????? + ???????? Suggested Homework: Study for quiz Section 5.2: 22, 25, 26, 32 Section 5.3: 4, 6, 8, 22, 29 th Math 340 Lecture – Introduction to Ordinary Differential Equations – April 27 , 2016 What We Covered: 1. Quiz over 4.5 and 5.1 2. Course Content – Chapter 5: The Laplace Transformation a. Section 5.3: The Inverse Laplace Transform i. So the whole idea here is that we want to go from F(s) – our solution – to f(t) – our equation. ii. One method we have is to use partial fractions 1. Most Laplace transforms are rational functions 2. Under the partial fractions we have two ways to proceed to the solution. We can either use the coefficient method or the substitution method. The coefficient methods has the advantage that it is straightforward and always works for any partial fraction problem. The substitution method is easier but only works when the denominator factors into simple linear factors. a. So which method should be used? Tis up to you, friend. The key idea is that we need to compute a number of coefficients, say N. To do so we need N equations. We get very simple equations if we substitute the roots of the denominator Suggested Homework: Section 5.3: 4, 6, 8, 22, 29 Math 340 Lecture – Introduction to Ordinary Differential Equations – April 29 , 2016 What We Covered: 1. Worksheet 11 a. I can’t go over the problems because it was assigned as homework, however we reviewed some it in class Suggested Homework: Worksheet 11

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