Introduction to Laplace Transform Methods Handout

Introduction to Laplace Transform Methods Handout -...

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1 Introduction to Laplace Transform Methods Why we care: Laplace Transform simplifies the solution of differential equations by allowing us to convert (1) Linear ODE into an algebraic equation in terms of Laplace variable s . (2) PDE into an ODE (with respect to original variable) Steps: 1) Transform our equation into Leplace Space 2) Integrate in Leplace Space 3) Transform the equation back from Leplace Space Definition: Given expression ) ( t f , its Laplace transform   ) ( ) ( s F t f L is defined as:       0 t dt t f e f L st F(s) (1) Simple Transforms: Utilizing the above definition, Laplace transforms for many familiar expressions have been obtained and tabulated (see attached) . A few examples are listed below: 1 f(t) s 1 dt e {1} 0 st L (2) at e f(t) a s 1 dt e dt e e } {e 0 a)t s ( 0 at st at L (3) at sin f(t) A A 0 a)t s ( 0 st dt e lim dt at sin e } at {sin L       A st A st A dt at e a s a at e 0 0 cos cos lim   0 cos 1 dt at e a s a st
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2   s F a s a 2 2 1 Solving for ) ( s F yields: 2 2 ) ( a s a s F (4) Key Rules: (A) Linearity of Laplace Transform. Given functions 1 f and 2 f whose Laplace transforms exist (t)} {f c (t)} f {c c (t)} f c (t) f {c 2 2 1 1 1 2 2 1 1 L L L (5) (B) Treatment of Derivatives (It is a theorem) Given continuous function f and ' f f(0) {f(t)} s } dt df { L L (6) (0) f' sf(0) {f(t)} s } dt f d { 2 2 2 L L (7) General Case: ... f(0) s {f(t)} s } {f 1 - n n (n) L L ) 0 ( 1) - (n 2) - (n f (0) sf (8)
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3 Example 1: Solving an initial value problem involving a linear homogeneous ordinary differential equation using Laplace transform. Given
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Introduction to Laplace Transform Methods Handout -...

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