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Unformatted text preview: © 2011 Zachary S Tseng 1 MATH 251 Work sheet / Things to know Chapter 6 1. The Laplace transforms The Laplace transform is an integral transformation that transforms a function of t (functions in tspace, or in engineering speak, in the time domain ) to a function of a second independent variable s (in sspcae, or the frequency domain ). It is defined by the following definite integral (note that this is an improper integral): L { f ( t )} = F ( s ) = F ( s ) is called the transform or Laplace transform of f ( t ). This operation is onetoone for any function f ( t ) that is continuous on (0, ∞ ). Ex.6.1.1 Find the Laplace transform of f ( t ) = t . Q : What is the Laplace transform of f ( t ) = 0? 2. Some properties of Laplace transforms Linearity  Laplace transform is linear L { C 1 f ( t ) + C 2 g ( t )} = The derivative of a Laplace transform F ′( s ) = © 2011 Zachary S Tseng 2 The Laplace transform of a derivative L { f ′( t )} = Therefore, L { f ″( t )} = L { f ′″( t )} = etc. 3. Solving initial value problems Before you proceed : review rules of partial fractions. The method of the Laplace Transform is a whole “new system” of solving linear differential equations algebraically . The system works essentially the same way regardless the specifics of each linear equation in question (it does not require a separate step such as the method of undetermined coefficients for a nonhomogeneous equation). What are the 3 stages of using the Laplace transform to solve a differential equation? © 2011 Zachary S Tseng 3 Ex. 6.3.1 ( Ex. 3.3.1.a ) y ″ + y ′  12 y = 0, y (0) = 0, y ′(0) = 2 Without any modification, a nonhomogeneous linear equation can also be solved using the 3¡step process: Ex. 6.3.2 y ″ + 5 y ′ + 6 y = 3 e 4 t , y (0) = 1, y ′(0) = 1 © 2011...
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 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Derivative, Partial Differential Equations, Continuous function, Zachary S Tseng

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