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set17 - $ Set 17 Laplace Transform and IVPs Part 2 Kyle A...

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a39 a38 a36 a37 Set 17: Laplace Transform and IVPs Part 2 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1
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a39 a38 a36 a37 Some Important Properties Let L{ y ( t ) } = Y ( s ) and L{ f i ( t ) } = F i ( s ) . (Table 6.2.1 in textbook has summary of many imporant properties) L{ α 1 f 1 ( t ) + · · · + α k f k ( t ) } = α 1 F 1 ( t ) + · · · + α k F k ( t ) L{ y } = sY ( s ) y (0) and L{ y ′′ } = s 2 Y ( s ) sy (0) y (0) L{− tf ( t ) } = F ( s ) and g ( t ) = integraldisplay t 0 f ( τ ) G ( s ) = F ( s ) s f ( t ) = f ( t + T ) , t 0 , T > 0 F ( s ) = integraltext T 0 f ( t ) dt 1 e sT 2
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a39 a38 a36 a37 Some Forcing Functions of Interest unit step (Heaviside) function for c 0 Dirac delta function pulse function square wave ramp loading saw tooth wave rectified sine wave 3
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a39 a38 a36 a37 Overview We will consider: Heaviside function, u c ( t ) , and L{ u c ( t ) } Other functions defined in terms of u c ( t ) and their transforms Dirac delta and its transform. 4
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a39 a38 a36 a37 Heaviside Function Unit step (Heaviside) function for c 0 u c ( t ) = 0 t < c 1 t c 5
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a39 a38 a36 a37 Heaviside Function -1 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 1.5 2 2.5 3 u 2 ( t ) for t 0 6
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a39 a38 a36 a37 Step Down via Heaviside -1 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 1.5 2 2.5 3 1 u 4 ( t ) for t 0 7
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a39 a38 a36 a37 Pulse Function f ( t ) = γ τ t τ 0 otherwise 8
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a39 a38 a36 a37 Pulse Function -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Pulse with γ = 1 and τ = 2 9
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a39 a38 a36 a37 Square Wave Simple square wave, e.g., 1 and 0 alternately infinitely each with width 1 .
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