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The Laplace Transform, , , , , , , , Syllabus, , , , Introduction, Definition of the Laplace Transform, Table of Elementary Laplace Transforms, Theorems on Important, Properties of Laplace Transformation, First Shifting. Theorem, Second Shifting Theorem, The Convolution Theorem,, Laplace Transform of an Integral, Laplace Transform of Derivatives, , , , , , , , SIE Lol hm el ol fot, , 6.1 _ Introduction, , The laplace transform is very important tool to the, engineer and scientist to find solutions to intial value, problems involving homogeneous and non homogeneous, differential equations alike., , The method can be extended to solve system of, differential equations, partial differential equations and, integral equations., , In this chapter we start with a definition of the laplace, transform and state some sufficient conditions for its, existence . We then derive its general properties and develop, a table of transform of some functions which are usually, encountered in the solution of differential equatins., , 6.1.1 Prerequisite, , 1, Gamma function : [0 = fe™*x"~' dx (n>0), 0, , 2 {nti =nfn, =n! if nis integer, , , , Je du(c) erf @) +erf.()=1, , “i!, , e~ sin (bt +c) dt=, , () erf.(f), , , , , , “cos (bt +c) dt=—~, a+b, , , , 7 J, 0, s| 7, 0, b, 9. DUIS [Rule I]: If (a) = fr (x, a) dx a, b are, a, , b, a fe, constants then, 40, = le [f (x, ce] dx., a, , Defin, Transform, , , , Syllabus Topic of the Laplace, , 6.2 _ Definition of Laplace Transform, , Definition : If f (t) is a function of t defined for all, positive values of t, then the Laplace transform of f (t), denoted by L [f (t)] is defined as, fe*r@at, 0, , Provided that integral exist. s is a parameter which, be a real or complex. et, , L{f ()] is a function of s, which is denote by f(s), , Lif@] =, , + LEO) = F@= fe*e@at, 0, There is one to one correspondence between f (t) and f (s), , f (© is called object function which is defined for

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6.2.1 Existence Condition, , t20, f(s) is the resultant function or image function., , The Laplace transform of function f(t) exist a, , Satisfies following conditions :, @_ f © is piecewise continuous on every, (ii) f(t) satisfies the following equality :, , finite interval., , [£(| <b e" for all t2 0. Where a, b are constants., Condition (ii) is known as exponential order., , Syllabus Topi Elem, eT lemme lat, , , , lala, i, , 6.3 Laplace Transform of Standard, , Functions, , SKE MNCUONS nies ee, , ® f@®=1, , We have, L[f(t)] = fetr@at, , La = fe"a@a =, , °, , @ f@®=e, , , , (it) f(t) =sinat, , sin@ =, , i, : elit _ gan, sinat = ——3,, , “By definition ,, , L(sin at) = J oe” sinatdt, 0, , ...( by definition), , (s>0), , «(by definition), , [ifs—a>0], , , , , , a, 2 L(sinat)= 2,7, , (v), , , , , , , , , , , , , ant one, , oo, J, 0, = oo, we fe*e™ dt — Jchoum dt, 0 0, , Lee, total, , £, , st+a, , , , f(t)=cos at, io, ie, e+e, ~ cos8 = 2, clit 4 ei, comet ey eae, By definition ,, L(cos at) = fe-“‘cos at dt