Note for Applied Mathematics - 2 - M-2 by Deepak

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ADD 9413003210 IN YOUR CLASSGROUP TO GET MORE FILES. APPLIED MATHEMATICS 7th Chapter Laplace Transform 3rd Semester, SBTE BIHAR (As Per New Syllabus Effective from 2016-2019 Batch) 7th Chapter Laplace Transform FOR STUDENTS For Free PDF of book ( as per new syllabus) whatsapp your following details to 9413003210 Name : College : Semester : You can study from this PDF. Only One chapter will be sent by whatsapp. Available all Subject of 3rd Semester As per new syllabus of SBTE Bihar FOR TEACHERS ONLY For Specimen Copy of Book of your subject whatsapp following details at 9460943210 Name : College : Subject : JHUNJHUNUWALA PUBLICATIONS Whatsapp 9413003210

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Add 9413003210 in your class whatsapp group to get more files. CHAP 1 7_LAPLACE TRANSFORM PAGE 1 CHAPTER 7 7_LAPLACE TRANSFORM 1.1 INTRODUCTION The Laplace transform is a very versatile tool to the engineer and scientist. It enables one to find solutions to initial-value problems involving homogeneous and non-homogeneous equations. The method can be extended to solve systems of differential equations, partial differential equations and integral equations. Its importance lies in the fact that its application is considerably easier than other available techniques. In this chapter, we will start with a definition of the transform and state some sufficient conditions for its existence. We then derive its general properties and develop a table of transforms of some functions which are usually encountered in the solution of linear differential equations. 1.2 DEFINITION OF LAPLACE TRANSFORM Let f ^ t h be a given function defined for all positive values of t , i.e. t $ 0 . Then, the Laplace transform of f ^ t h is defined as the function F ^s h such that L 6f ^ t h@ = F ^s h = #e 3-st 0 f ^ t h dt ...(1) where L is referred to as the Laplace transform operator. Generally, the transform will exist for more than one real value of s , and hence L 6f ^ t h@ defines a function of s when it exists. 1.3 SUFFICIENT CONDITION FOR EXISTENCE While finding the Laplace transforms of elementary function, it can be noticed that the integral exists under certain conditions, such as s > 0 as s > a etc. In general, the function f (t) must satisfy the following conditions for the existence of the Laplace transform. 1. The function f (t) must be piece-wise continuous or sectionally continuous in any limited interval 0 < a # t # b . 2. The function f (t) is of exponential order. Piece-wise Continuous Function A function f (t) is said to be piece-wise (or sectionally) continuous over the closed interval [a , b] if it is defined on that interval and is such that the interval can be divided into a finite number of subintervals, in each of which f (t) is continuous and has both right and left hand limits at every end point of the subinterval. e.g. : 1. The function f (t) = t2 , 0 < t < 5 = 2t + 3 , t > 5 is sectionally continuous for t > 0 . 2. The function f (t) = 1t is not sectionally continuous in any interval containing t = 0. Functions of Exponential Order A function f (t) is said to be of exponential order a if,

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Add 9413003210 in your class whatsapp group to get more files. PAGE 2 7_LAPLACE TRANSFORM CHAP 1 lim e-at f (t) = a finite quantity t"3 i.e., for a given positive number T , there exists a real number M > 0 such that, e-at f (t) < M , 6t $ T or f (t) < Meat , 6t $ T For example, f (t) = t2 , sin at , eat etc. are all of exponential order and also continuous. But f (t) = et is not of exponential order and as such its Laplace transform does not exist. 2 1.4 LAPLACE TRANSFORM OF SOME ELEMENTARY FUNCTIONS We have just discussed that Eq(1.2) is the generalised form of Laplace transformation for the causal function that assumes zero value for t < 0 . Using the transformation, let us obtain the Laplace transforms of some elementary functions (causal functions). 1.4.1 Unit Impulse Function Unit impulse function is defined as 0, t ! 0 d^t h = * 3, t = 0 So, the Laplace transform of unit impulse function is given by L 6d^ t h@ = #e 3-st 0 d^ t h dt = lim 6e-st d^ t h@ t"0 =1 1.4.2 Unit Step Function Unit step function is defined as 1, u^t h = * 0, t$0 t<0 So, the Laplace transform of unit step function is given by L 61@ = # ^e 3 -st 0 -st 3 h^1 h dt = :e- s D = :0 - -1s D 0 = 1 , for s > 0 s 1.4.3 Exponential Function Let the exponential function, f ^ t h = eat Its Laplace transform is obtained as L 6eat@ = = # 0 3-st at e e dt = # 3 - s-a t ^ h 0 1 , for s > a s-a e dt = e-^s - a ht -^s - a h 3 0

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Add 9413003210 in your class whatsapp group to get more files. CHAP 1 1.4.4 7_LAPLACE TRANSFORM PAGE 3 Trigonometric Function Let the sine function, f ^ t h = sin at Its Laplace transform is obtained as iat -iat L 6sin at @ = L ;e - e E 2i iat e-iat E ;sin at = e 2i = 1 8L ^eiat h - L ^e-iat hB 2i = 1 : 1 - 1 D = 1 c 22ia 2 m :L ^eiat h = 1 D 2i s - ia s + ia 2i s + a s - ia = 2 a 2 , for s > 0 s +a Again, let the cosine function, f ^ t h = cos at Its Laplace transform is obtained as iat -iat L 6cos at @ = L ;e + e E 2 iat -iat ;cos at = e +2 e E = 1 8L ^eiat h + L ^e-iat hB 2 = 1 : 1 + 1 D = 1 c 2 2s 2 m 2 s - ia s + ia 2 s +a = 2 s 2 , for s > 0 s +a 1.4.5 1 iat :L ^e h = s - ia D Hyperbolic Function Let the sine hyperbolic function, f ^ t h = sinh at Its Laplace transform is obtained as at -at L 6sinh at @ = L ;e - e E 2 at -at ;sinh at = e -2 e E = 1 8L ^eat h - L ^e-at hB 2 = 1 : 1 - 1 D = 1 b 2 2a 2 l 2 s-a s+a 2 s -a = 2 a 2 , for s > a s -a 1 at :L ^e h = s - a D Again, let the cosine hyperbolic function, f ^ t h = cosh at Its Laplace transform is obtained as at -at L 6cosh at @ = L ;e + e E 2 at -at ;cosh at = e +2 e E = 1 8L ^eat h + L ^e-at hB 2 = 1 : 1 + 1 D = 1 b 2 2s 2 l 2 s-a s+a 2 s -a 1 at :L ^e h = s - a D