notes 6-2 - terms, treat each simpler term separately, then...

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SECTION 6.2 USING THE LAPLACE TRANSFORM A CALCULATION. Suppose f ( t ) does not grow too fast as t → ∞ and that f 0 ( t ) is continuous for t 0. Find the Laplace transform of f 0 ( t ). MORE CALCULATION. What about the Laplace transform of f 00 ( t )? Higher deriva- tives? This pattern continues for all derivatives and works as long as the highest derivative is piecewise continuous and all functions involved grow no faster than Ke at as t → ∞ . All our functions will be like this!
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EXAMPLE. Find the Laplace transform Y ( s ) of the solution of the initial value problem y 00 - 2 y 0 + 2 y = e - t , y (0) = 0 , y 0 (0) = 1 .
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Given Y ( s ), we need to find the function φ ( t ) whose Laplace transform is Y ( s ). We sometimes use the notation L - 1 { Y ( s ) } to denote this inverse transform of Y ( s ). Could there be several such φ ( t )’s? Answer: NO. Further, the inverse Laplace transform is also a linear operator, which means we can split Y ( s ) into sums of constant multiples of simpler
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Unformatted text preview: terms, treat each simpler term separately, then recombine. There is a general theory of the inverse Laplace transform and of course a method of calculation, but its beyond us. All we need is presented in the table on Page 319. This table will be provided when needed for exams and quizzes. To nd L-1 { Y ( s ) } we need to write Y ( s ) as a sum of terms that match entries in the middle column of the table, and L-1 { Y ( s ) } will then be the sum of the corresponding terms in the left column of the table. EXAMPLE. Find the inverse Laplace transform of 8 s 2-4 s + 12 s ( s 2 + 4) . EXAMPLE. Find the solution of the initial value problem y 00-2 y + 2 y = e-t , y (0) = 0 , y (0) = 1 . HOMEWORK: SECTION 6.2...
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notes 6-2 - terms, treat each simpler term separately, then...

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