# 9 - Second Translation Theorem: If ) ( ) ( t f s F L and a...

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MATH2860U: Chapter 7 cont… 1 THE LAPLACE TRANSFORM cont… Operational Properties I (Section 7.3 of Zill and Cullen, pg. 270) Recall: Last week, you did some Laplace with translations and also solved some initial value problems using the Laplace transform. E.g. Question: In engineering applications, one often encounters discontinuous forcing functions…how can we solve differential equations in which the nonhomogeneous term is a piecewise defined function? To help us deal with jump discontinuities, we introduce a special function. Definition: The unit step function (also known as the Heaviside function ) is defined by a t a t a t 0 , 1 , 0 ) ( U Examples:

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MATH2860U: Chapter 7 cont… 2 Question: What’s the Laplace transform of the unit step function?
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Unformatted text preview: Second Translation Theorem: If ) ( ) ( t f s F L and a , then ) ( ) ( ) ( s F e a t a t f as U Conversely, if ) ( ) ( 1 s F t f , then ) ( ) ( ) ( 1 a t a t f s F e as U Example: Example: Find 5 8 1 s e s Example: Find 2 2 2 1 s s e s Now let’s put everything together now and finally solve an ODE with a discontinuous forcing function. MATH2860U: Chapter 7 cont… 3 Example: Solve ) ( 3 2 t g y y y where 2 2 , 5 , 1 ) ( t t t g subject to ) ( , ) ( y y ....
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## This note was uploaded on 11/10/2010 for the course DEFFERENTI 2080 taught by Professor Kidnan during the Spring '10 term at UOIT.

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9 - Second Translation Theorem: If ) ( ) ( t f s F L and a...

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