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 : ADVANED ENGINEERING MAHEMATICS – ERWIN KREYSZIG [8 th EDITION]

3. MATHEMATICS-II Differentiation and Integration of Laplace Transforms Lecture : 6 DEPARTMENT OF MATHEMATICS, CVRCE

4. OUTLINES  Differentiation of transforms  Integration of transforms  Linear equations with variable coefficients  Problems based on these topics OUTLINES  Differentiation of transforms  Integration of transforms  Linear equations with variable coefficients  Problems based on these topics DEPARTMENT OF MATHEMATICS, CVRCE

5. DIFFERENTIATION OF TRANSFORMS THEOREM: Let f(t) be a function whose laplace transform exists and then PROOF: By definition of Laplace transform we have

6. DIFFERENTIATION OF TRANSFORMS [generalisation] THEOREM: Let f(t) be a function whose laplace transform exists and

7. Solution: 1. Find DIFFERENTIATION OF TRANSFORMS [problems]

8. 2. Find Solution : DIFFERENTIATION OF TRANSFORMS [problems]

9. SOLUTION OF PROBLEM - 2

10. 3. Find Solution : DIFFERENTIATION OF TRANSFORMS [problems]

11. Find the Laplace transformation using differentiation Assignment DIFFERENTIATION OF TRANSFORMS [problems]

12. INTEGRAATION OF TRANSFORMS THEOREM: Let f(t) be a function whose laplace transform exists and then PROOF: By definition of Laplace transform we have

13. INTEGRAATION OF TRANSFORMS [By altering the order of integration]

14. IMPORTANT RESULTS OF CALCULUS TO BE REMEMBERED

15. Example1: Find the inverse of using integration and differentiation of laplace trasform. Solution: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS METHOD-i: [BY INTEGRATION OF LAPLACE TRANSFORM]:

16. PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS METHOD-i: [BY DERIVATIVE OF LAPLACE TRANSFORM]:

17. Example2: Find the inverse of using differentiation and integration of laplace transform. Solution: METHOD-i: [BY DERIVATIVE OF LAPLACE TRANSFORM]: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

18. [By derivative of Laplace transform] PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

19. METHOD-ii: [BY INTEGRATION OF LAPLACE TRANSFORM]: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

20. Example 3: Find the inverse of using differentiation and integration of laplace transform. METHOD-i: [BY INTEGRATION OF LAPLACE TRANSFORM]: Solution: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

23. [The details of the computation may be seen in the last two slides.] METHOD-ii: [BY DERIVATIVE OF LAPLACE TRANSFORM]: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

24. Example 4: Find the inverse of using differentiation and integration of laplace transform. METHOD-i: [BY INTEGRATION OF LAPLACE TRANSFORM]: Solution: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

26. METHOD-ii: [BY DERIVATIVE OF LAPLACE TRANSFORM]: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

27. Example 5: Find the inverse of using differentiation and integration of laplace transform. METHOD-i: [BY DERIVATIVE OF LAPLACE TRANSFORM]: Solution: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

29. METHOD-ii: [BY INTEGRATION OF LAPLACE TRANSFORM]: PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS

30. Assignment Find the inverse using differentiation or integration PROBLEMS INVOLVING DERIVATIVE AND INTEGRAATION OF TRANSFORMS