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 FOR BTECH SECOND SEMESTER COURSE [COMMON TO ALL BRANCHES OF ENGINEERING] DEPARTMENT OF MATHEMATICS, CVRCE TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION]

3. MATHEMATICS - II LECTURE :4-5 - Laplace Transforms of Unit Step Function [Second Shifting Theorem] and Dirac Delta Function, Solution of Some Initial Value Problems by Laplace Transforms and Modeling Problems of Electric circuit [LC, RC, LR circuits] Problems. [Chapter – 5.3] DEPARTMENT OF MATHEMATICS, CVRCE

4. 3/23/1749 - 3/2/1827 Pierre-Simon Laplace

5. DEPARTMENT OF MATHEMATICS, CVRCE LAYOUT OF LECTURE LAPLACE TRANSFORM OF UNIT STEP FUNCTION SECOND SHIFTING THEOREM LAPLACE TRANSFORM OF IUNIT IMPULSE FUNCTIONS APPLICATIONS IN SOLVING INITIAL VALUE PROBLEMS INVOLVING LINEAR DIFFERENTIAL EQUATIONS APPLICATIONS TO LC, RC, AND RL ELECTRIC CIRCUITS

6. DEPARTMENT OF MATHEMATICS, CVRCE Definition: The unit step function is defined as follows: UNIT STEP FUNCTION The unit step function is also called the Heaviside function. Unit Step function u(t-a) t 1 u(t-a) a

7. DEPARTMENT OF MATHEMATICS, CVRCE UNIT STEP FUNCTION

8. DEPARTMENT OF MATHEMATICS, CVRCE Theorem: If Then Proof: Given In other words, SECOND SHIFTING THEOREM

9. DEPARTMENT OF MATHEMATICS, CVRCE where t – a = u SECOND SHIFTING THEOREM

10. PROBLEMS INVOLVING UNIT STEP FUNCTION Example 1. Find Laplace transforms of Solution : Given function is Therefore,

11. PROBLEMS INVOLVING UNIT STEP FUNCTION By second shifting theorem we have

12. PROBLEMS INVOLVING UNIT STEP FUNCTION Example 2. Find Laplace transforms of Solution:

13. PROBLEMS INVOLVING UNIT STEP FUNCTION Example 3. Find inverse Laplace transforms of Solution:

14. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Example 4. Find inverse Laplace transforms of Solution:

15. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION

16. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Example 5. Using Laplace transformation solve the following initial value problem. Solution: Given differential equation is

17. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Taking Laplace transform of (1) we get

18. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Using (2) and (3) in the above equation we get

19. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Taking inverse Laplace transform of the above equation we get

20. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION is the desired solution.

21. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Example 6. Using Laplace transformation solve the following initial value problem. Solution : Given differential equation is

22. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Taking Laplace transform of (1) we get

23. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Using (2) and (3) in the above equation we get

24. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Therefore, the required solution is

25. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION

26. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION

27. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION

28. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING UNIT STEP FUNCTION Therefore from (5) we get the required solution is which is given by

29. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Example 7. Using Laplace transform determine current in the LR circuit with L = 1henry, R = 100ohms, initial current is 0, and the voltage given by Solution: Equation governing LR ciruit with current i(t) ampere, voltage v(t), resistance R ohm and inductance L henry is given by Here L = 1henry, R = 100ohms, i(0)=0, and

30. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Taking laplace transform of (2) we get

31. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS]

32. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Where A, B and C are constants to be evaluated as follows. Equating coeficient of each power of s in (4) we get. Solving the above equations we get we get.

33. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Required current i(t) is given by

34. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Example 8. Using Laplace transform determine current in the LC circuit with L = 1henry, C = 1 Farad, 0 initial current and charge, and the voltage given by Solution: Equation governing LC ciruit with current i(t) ampere, voltage v(t), resistance R ohm and capacitance C Farad is given by Here C = 1henry, C = 1Farad, i(0)=0, i ’ (0)=0 and

35. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS]

36. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] According to the question v(0) = 0 Taking Laplace transform of (2), we get

37. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Where A, B and C are constants to be evaluated as follows.

38. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Equating coeficient of each power of s in (4) we get. Solving the above equations we get we get. Required current i(t) is given by

39. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS]

41. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Required current is

42. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Example 9. Using Laplace transform determine current in the RC circuit C= 0.1Farad, R = 100ohms, 0 initial current and charge, and the voltage given by Solution: Equation governing RC circuit with current i(t) ampere, voltage v(t), Capacitance C Farad and resistance R Ohm is given by Here C = 0.1Farad, R = 100ohms, i(0)=0, and

43. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Taking laplace transform of (2) we get

44. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS OF ELECTRIC CIRCUIT [LC, RC, LR CIRCUITS] Required current i(t) is given by

45. UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] a+  1/  Area = 1 a t The impulse of   (t-a), say I  is given by

46. DEPARTMENT OF MATHEMATICS, CVRCE UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] Using unit step function we get Laplace transform of is given by

47. DEPARTMENT OF MATHEMATICS, CVRCE UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] The unit impulse function (Dirac Delta function) is defined as the limiting form of the function as   0

48. DEPARTMENT OF MATHEMATICS, CVRCE UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION]

49. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] Example 10: Using Laplace transformation solve the following differential equation Solution: The given initial value problem is

50. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] Taking Laplace transform of (1), we get

51. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION]

52. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] Equating coeficient of each power of s in (4) we get. Solving the above equations we get we get.

53. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION]

54. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] The required solution is given by

55. DEPARTMENT OF MATHEMATICS, CVRCE Example 11: Using Laplace transformation solve the following differential equation PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] Solution: The given initial value problem is

56. DEPARTMENT OF MATHEMATICS, CVRCE Taking Laplace transform of (1), we get PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION]

57. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION]

58. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION]

59. DEPARTMENT OF MATHEMATICS, CVRCE Equating coeficient of each power of s in (4) we get. Solving the above equations we get we get. PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION]

60. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS BASSED ON UNIT IMPULSE FUNCTION [DIRAC DELTA FUNCTION] Required solution is