Chapter 4 Lecture Notes-Laplace Transforms 2

# Chapter 4 Lecture Notes-Laplace Transforms 2 - McGill...

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McGill University Math 263: Differential Equations and Linear Algebra CHAPTER 4: LAPLACE TRANSFORMS (II) 1 Solve IVP of DE’s with Laplace Trans- form Method In this lecture we will, by using examples, show how to use Laplace transforms in solving differential equations with con- stant coefﬁcients. 1.1 Example 1 Consider the initial value problem y ′′ + y + y = sin( t ) , y (0) = 1 , y (0) = - 1 . 0-0

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Let Y ( s ) = L{ y ( t ) } , we have L{ y ( t ) } = sY ( s ) - y (0) = sY ( s ) - 1 , L{ y ′′ ( t ) } = s 2 Y ( s ) - sy (0) - y (0) = s 2 Y ( s ) - s + 1 . Taking Laplace transforms of the DE, we get ( s 2 + s + 1) Y ( s ) - s = 1 s 2 + 1 . Step 2 Solving for Y ( s ) , we get Y ( s ) = s s 2 + s + 1 + 1 ( s 2 + s + 1)( s 2 + 1) . Step 3 Finding the inverse laplace transform. y ( t ) = L 1 { Y ( s ) } = L 1 ± s s 2 + s +1 ² + L 1 ± 1 ( s 2 + s +1)( s 2 +1) ² . 0-1
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## This note was uploaded on 12/01/2010 for the course MATH 263 taught by Professor Sidneytrudeau during the Fall '09 term at McGill.

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Chapter 4 Lecture Notes-Laplace Transforms 2 - McGill...

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