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Section 7.2 - Differential Equations Math 308 Spring 2008...

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Differential Equations Texas A&M University Math 308, Spring 2008 c circlecopyrt 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 F ( s ) = integraldisplay 0 e - st f ( t )d t = lim N →∞ integraldisplay N 0 e - st f ( t )d t 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 ) = e 3 x . f ( x ) = 3 + e - 2 x . f ( x ) = sin x Theorem: Let f 1 and f 2 be 2 functions whose Laplace transforms exist for s > α and let c 1 and c 2 be two real constants. Then for s > α , L { f 1 + f 2 } = L { f 1 } + L { f 2 } L { c 1 f 1 + c 2 f 2 } = c 1 L { f 1 } + c 2 L { f 2 } Exercise 2. Let f ( x ) = braceleftbigg 3 if x [0 , 10] e x - 10 if x (10 , ) . Find the Laplace transform of f . 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
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