6.1 - Math 334 Lecture #23 6.1: The Laplace Transform An...

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Math 334 Lecture #23 § 6.1: The Laplace Transform An Idea. Is there an invertible linear “operator” L{ y ( t ) } = Y ( s ) that tranforms an IVP ± y 00 ( t ) + ay 0 ( t ) + by ( t ) = 0 , y (0) = y 0 , y 0 (0) = y 0 0 , into an algebraic equation like ( s 2 + as + b ) Y ( s ) - ( as + b ) y 0 - ay 0 0 = 0? [This asks if it is possible to transform the operation of differentiation with respect to the variable t into the operation of multiplication by a another variable s .] If so, this algebraic equation is easily solved for the transform of the solution of the IVP, Y ( s ) = ( as + b ) y 0 + ay 0 0 s 2 + as + b , and then applying the inverse of the operator to Y ( s ) gives the solution of the IVP: y ( t ) = L - 1 { Y ( s ) } . An Operator. The Laplace transform of a function f ( t ) with domain t 0 is the function F ( s ) defined by the improper integral F ( s ) = L{ f ( t ) } = Z 0 e - st f ( t ) dt = lim A →∞ Z A 0 e - st f ( t ) dt whose domain consists of those values of s for which the improper integral converges. Which functions have a Laplace transform? Under what conditions on
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6.1 - Math 334 Lecture #23 6.1: The Laplace Transform An...

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