1. DEPARTMENT OF MATHEMATI CS [ YEAR OF ESTABLISHMENT – 1997 ] DEPARTMENT OF MATHEMATICS, CVRCE

2. MATHEMATICS - II ● LAPLACE TRANSFORMS ● FOURIER SERIES ● FOURIER TRANSFORMS ● VECTOR DIFFERENTIAL CALCULUS ● VECTOR INTEGRAL CALCULUS ● LINE, DOUBLE, SURFACE, VOLUME INTEGRALS ● BETA AND GAMMA FUNCTIONS ● LAPLACE TRANSFORMS ● FOURIER SERIES ● FOURIER TRANSFORMS ● VECTOR DIFFERENTIAL CALCULUS ● VECTOR INTEGRAL CALCULUS ● LINE, DOUBLE, SURFACE, VOLUME INTEGRALS ● BETA AND GAMMA FUNCTIONS FOR BTECH SECOND SEMESTER COURSE [COMMON TO ALL BRANCHES OF ENGINEERING] DEPARTMENT OF MATHEMATICS, CVRCE TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS – ERWIN KRYSZIG [8 th EDITION]

3. MATHEMATICS-II Laplace Transforms Using Partial Fractions Lecture : 9 DEPARTMENT OF MATHEMATICS, CVRCE

4. Inverse Laplace Transforms Using Partial Fractions Let the Laplace transform of a function Y(s) be a fraction of the form with degree of F(s) is less than that of G(s). G(s) consists of product of factors of the form, say (s-a).These factors may be one of the following types: i] Unrepated factors ii] Repeated factors iii] Unrepeated complex factors iv] Repeated complex factors

5. Laplace Transforms Using Partial Fractions With the degree of numerator is less than that of the denominator Case(I) Equating the coefficient of each power of s in (2), we get n equations involving the coefficients A 1, A 2,..., A n we get n- linear equation solving which we get the values of A 1, A 2,..., A n

6. Laplace Transforms Using Partial Fractions With the degree of numerator is less than that of the denominator Case(II) Equating the coefficient of each power of s in (4), we get n equations involving the coefficients A 1, A 2,..., A n we get n- linear equation solving which we get the values of A 1, A 2,..., A n

7. Laplace Transforms Using Partial Fractions Case(III) With the degree of numerator is less than that of the denominator Equating the coefficient of each power of s in (4), we get n equations involving the coefficients A 1, A 2,..., A n we get n- linear equation solving which we get the values of A 1, A 2,..., A n

8. Laplace Transforms Using Partial Fractions Case(IV) With the degree of numerator is less than that of the denominator Equating the coefficient of each power of s in (4), we get n equations involving the coefficients A 1, A 2,..., A n we get n- linear equation solving which we get the values of A 1, A 2,..., A n

9. Laplace Transforms Using Partial Fractions Case(V) With the degree of numerator is less than that of the denominator

10. Laplace Transforms Using Partial Fractions

11. Equating the coefficient of each power of s in (4), we get n equations involving the coefficients A 1, A 2,..., A n we get n- linear equation solving which we get the values of A 1, A 2,..., A n

12. Problems Involving Laplace Transforms Using Partial Fractions 1. Find the inverse Laplace transform by partial fraction Solution : Equating the coefficient of each power of s in (1), we get

13. Problems Involving Laplace Transforms Using Partial Fractions On solving above equations we get

14. Problems Involving Laplace Transforms Using Partial Fractions 2. Find the inverse Laplace transform by partial fraction Solution : Equating the coefficient of each power of s in (1), we get

15. Problems Involving Laplace Transforms Using Partial Fractions On solving above equations we get

16. Problems Involving Laplace Transforms Using Partial Fractions 3. Find the inverse Laplace transform by partial fraction Solution :

17. Problems Involving Laplace Transforms Using Partial Fractions Equating the coefficient of each power of s in (1), we get

18. Problems Involving Laplace Transforms Using Partial Fractions

19. On solving equations (2) and (11) we get From equation (9) we get From equation (3) we get From equation (6) we get

20. Problems Involving Laplace Transforms Using Partial Fractions

21. 4. Find the inverse Laplace transform by partial fraction Problems Involving Laplace Transforms Using Partial Fractions Solution :

22. Problems Involving Laplace Transforms Using Partial Fractions Equating the coefficient of each power of s in (1), we get

23. Problems Involving Laplace Transforms Using Partial Fractions On solving equations (2) and (6) we get On solving equations (5) and (7) we get

24. Problems Involving Laplace Transforms Using Partial Fractions

25. 5. Show that Solution :

26. Problems Involving Laplace Transforms Using Partial Fractions

27. Equating the coefficient of each power of s in (1(a)), we get

28. Problems Involving Laplace Transforms Using Partial Fractions Applying equations (1) and (4) in (2) and (3) we get

29. Equations (1), (4), (5), and (6), give Problems Involving Laplace Transforms Using Partial Fractions

31. 6. Show that

32. Problems Involving Laplace Transforms Using Partial Fractions

33. Equating the coefficient of each power of s in (1(a)), we get

34. Applying equations (1) and (4) in (2) and (3) we get Problems Involving Laplace Transforms Using Partial Fractions

35. Equations (1), (4), (5), and (6), give Problems Involving Laplace Transforms Using Partial Fractions

39. 5. Show that Solution :

40. 6. Show that Solution :

41. Assignment Find the inverse Laplace transform using partial fraction