Section 7.2

Section 7.2 - Differential Equations Math 308 Spring 2008 Texas A&M University c F Dos Reis Section 7-2 Definition Let f be a function on[0 The

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Unformatted text preview: Differential Equations Math 308, Spring 2008 Texas A&M University c F. Dos Reis Section 7-2 Definition: Let f be a function on [0, ). The Laplace transform of f is the function F defined by the integral N F (s) = 0 e-st f (t)dt = lim N 0 e-st f (t)dt The domain of F is all the values of s for which the integral exists. The Laplace transform of f is also written L {f }. Exercise 1. Find the Laplace transforms of the following functions and give their domains. f (x) = 1. f (x) = e3 x. f (x) = 3 + e-2x . f (x) = sin x Theorem: Let f1 and f2 be 2 functions whose Laplace transforms exist for s > and let c1 and c2 be two real constants. Then for s > , L {f1 + f2 } = L {f1 } + L {f2 } L {c1 f1 + c2 f2 } = c1 L {f1 } + c2 L {f2 } Exercise 2. Let f (x) = 3 ex-10 if x [0, 10] . Find the Laplace transform of f . if x (10, ) Definition: A function f is said to be piecewise continuous on a finite interval [a, b] if f (t) is continous at every point in [a, b] except possibly for a finite number of points at which f has a jump discontinuity. A function f is said to be piecewise continuous on [0, ) if f is piecewise continuous on [0, N ] for all N > 0. Definition: A function f is said to be of exponential order if there exist positive constants T and M such that for all t > T , |f (t)| M et . Exercise 3. Which of the following functions are of exponential order? 1 Differential Equations Math 308, Spring 2008 e3x-1 sin 2x ex x3 ex 2 Texas A&M University c F. Dos Reis e-x 2 Theoerem: If f is piecewise continuous on [0, ) and of exponential order then L {f } (s) exists for all s > . Table of Laplace transforms: f (t) 1 eax tn n = 1, 2, sin bt cos bt eat tn n = 1, 2, eat sin bt eat cos bt F (s) 1 s 1 s-a n! sn+1 s2 s2 b + b2 s + b2 n! (s - a)n+1 b (s - a)2 + b2 s-a (s - a)2 + b2 Exercise 4. Find the Laplace transforms of f (t) = 6 e-3t -t2 + e5 t cos 2t 2 ...
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This note was uploaded on 06/24/2009 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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Section 7.2 - Differential Equations Math 308 Spring 2008 Texas A&M University c F Dos Reis Section 7-2 Definition Let f be a function on[0 The

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