δ t t dt 1 if t a b if t 6 a b The Laplace transform of δ t t follows easily L

# Δ t t dt 1 if t a b if t 6 a b the laplace transform

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δ ( t - t 0 ) dt = 1 , if t 0 [ a, b ) , 0 , if t 0 6∈ [ a, b ) . The Laplace transform of δ ( t - t 0 ) follows easily: L [ δ ( t - t 0 )] = Z 0 e - st δ ( t - t 0 ) dt = e - st 0 Note: L [ δ ( t )] = 1. The delta function is the symbolic derivative of the Heaviside function , so δ ( t - t 0 ) = u 0 ( t - t 0 ) This is rigorously true in the theory of generalized functions or distributions Joseph M. Mahaffy, h [email protected] i Lecture Notes – Laplace Transforms: Part B — (33/35) Inverse Laplace Transforms Special Functions Heaviside or Step function Periodic functions Impulse or δ Function Example for δ ( t - t 0 ) 1 Example: Consider the initial value problem: y 00 + 2 y 0 + 2 y = t π δ ( t - π ) , y (0) = 0 , y 0 (0) = 1 The Laplace transform of the forcing function is F ( s ) = Z 0 e - st t π δ ( t - π ) dt = e - πs It follows that the Laplace transform of the IVP is s 2 Y ( s ) - 1 + 2 sY ( s ) + 2 Y ( s ) = e - πs , so Y ( s ) = 1 + e - πs ( s + 1) 2 + 1 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Laplace Transforms: Part B — (34/35) Inverse Laplace Transforms Special Functions Heaviside or Step function Periodic functions Impulse or δ Function Example for δ ( t - t 0 ) 2 Example: Since Y ( s ) = 1+ e - πs ( s +1) 2 +1 , the inverse Laplace transform satisfies: y ( t ) = e - t sin( t ) + u π ( t ) e - ( t - π ) sin( t - π ) 0 1 2 3 4 5 6 7 8 9 10 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 t y ( t ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Laplace Transforms: Part B — (35/35)
• Fall '08
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