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

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]

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

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

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

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

Solution: 1. Find DIFFERENTIATION OF TRANSFORMS [problems]

2. Find Solution : DIFFERENTIATION OF TRANSFORMS [problems]

SOLUTION OF PROBLEM - 2

3. Find Solution : DIFFERENTIATION OF TRANSFORMS [problems]

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

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

INTEGRAATION OF TRANSFORMS [By altering the order of integration]

IMPORTANT RESULTS OF CALCULUS TO BE REMEMBERED

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]:

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

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

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

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

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

[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

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

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

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

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

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