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

This preview shows page 9 out of 9 pages.

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)
Image of page 9
  • Fall '08
  • staff

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes