This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Differential Equations Math 308, Spring 2008 Texas A&M University
c F. Dos Reis Section 72
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 est f (t)dt = lim N 0 est 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 + e2x . 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 ex10 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 e3x1 sin 2x ex x3 ex
2 Texas A&M University
c F. Dos Reis ex 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 sa n! sn+1 s2 s2 b + b2 s + b2 n! (s  a)n+1 b (s  a)2 + b2 sa (s  a)2 + b2 Exercise 4. Find the Laplace transforms of f (t) = 6 e3t t2 + e5 t cos 2t 2 ...
View
Full
Document
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.
 Spring '08
 comech
 Differential Equations, Equations

Click to edit the document details