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The Laplace Transform Let f(x) be defined for 0≤x<∞ and let s denote an arbitrary real variable. The Laplace transform of f(x) designated by either £{f(x)} or F(s), is for all values of s for which the improper integral converges.

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Convergence occurs when the limit exists. If the limit does not exist, the improper integral diverges and f(x) has no Laplace transform.

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When evaluating the integral in The variable s is treated as a constant because the integration is with respect to x.

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EXAMPLE Determine whether the improper integral converges.

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EXAMPLE Determine whether the improper integral converges.

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EXAMPLE Determine those values of s for which the improper integral converges.

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Laplace Transforms for a Number of Elementary Functions Find the Laplace Transform of f(x)=1

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Laplace Transforms for a Number of Elementary Functions Find the Laplace Transform of f(x)=x 2

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Laplace Transforms for a Number of Elementary Functions Find £{e ax }

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Laplace Transforms for a Number of Elementary Functions Additional transforms are given in Appendix A.

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Properties of Laplace Transforms Property 1. If then for any two constants C 1 and C 2

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Properties of Laplace Transforms Property 2. If then for any constant a

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Properties of Laplace Transforms Property 3. If then for any positive integer n

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Properties of Laplace Transforms Property 4. If then if exists, then

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Properties of Laplace Transforms Property 5. If then

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Properties of Laplace Transforms Property 6. If f(x) is periodic with period w, that is, f(x+w)=f(x), then

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Functions of Other Independent Variables For consistency only, the definition of the Laplace transform and its properties, the above equations are presented for functions of x. They are equally applicable for functions of any independent variable and are generated by replacing the variable x in the above equations by any variable of interest. In particular, the Laplace Transform of a function of t is

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Inverse Laplace Transforms An inverse Laplace transform of F(s), designated by £ -1 {F(s)} is another function f(x) having the property that £ {f(x)} =F(s). This presumes that the independent variable of interest is x. If the independent variable of interest is t instead, then an inverse Laplace transform of F(s) is f(t) where £ {f(t)} =F(s).

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The simplest technique for identifying inverse Laplace transforms is torecognize them, either from memory or from a table such as Appendix A. If F(s) is not a recognize form, then occasionally it can be transformed into such a form by algebraicmanipulation. Observe from Appendix A that almost all Laplace transforms are quotients. The recommended procedure is to first convert the denominator to a form that appears in Appendix A and then the numerator. Inverse Laplace Transforms

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Manipulating Denominators: The method of completing the square converts a quadratic polynomial into the sum of squares, a form that appears in many of the denominators in Appendix A. In particular, for the quadratic as 2 +bs+c, where a,b,and c denote constants. Inverse Laplace Transforms

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The method of partial fractions transforms a function of the form a(s)/b(s), where both a(s) and b(s) are polynomials in s, into the sum of other fractions such that the denominator of each new fraction is either a first degree or a quadratic polynomial raised to some power. The method requires only 1.The degree of a(s) be less than the degree of b(s) and 2. b(s) be factored into the product of distinct linear and quadratic polynomials raised to various powers.

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The method is carried out as follows.

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Manipulating Numerators: A factor s-a in the numerator may be written in terms of the factor s-b, where both a and b are constants, through the identity s-a=(s-b)+(b-a). The multiplicative constant a in the numerator may be written explicitly in terms of the multiplicative constant b through the identity a=(a/b)(b)

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Both identities generate recognizable inverse Laplace transforms when they are combined with: If the inverse Laplace transforms of two functions F(s) and G(s) exist, then for any constants C 1 and C 2.

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Denote £{y(x)} by Y(s). Then under very broad conditions, the Laplace transform of the nth- derivative (n=1,2,3,…) of y(x) is Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transforms

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If the initial conditions on y(x) at x=0 are given by y(0)=C 0, y’(0)=C 1,……….. y (n-1) (0)=C n-1, then For the special cases of n=1 and n=2

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Laplace transforms are used to solve initial-value problems given by the nth-order linear differential equation with constant coefficients Together with the initial conditions specified y(0)=C 0, y’(0)=C 1,……….. y (n-1) (0)=C n-1 Solutions of Differential Equations

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First, take the Laplace transform of both sides of the differential equation Thereby obtaining an algebraic equation for Y(s). Then solve for Y(s) algebraically, and finally take inverse Laplace transforms to obtain

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Example Solve y’-5y=0 ; y(0)=2 Solvey’+y=sinx ; y(0)=1 Solve y’’+4y=0 ; y(0)=2, y’(0)=2 Solvey’’-3y’+4y=0 ; y(0)=1, y’(0)=5

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