SRI VENKATESWARA COLLEGE OF ENGINEERING

DEPARTMENT OF APPLIED MATHEMATICS

WORK SHEET- MA6251- MATHEMATICS – II

UNIT III–LAPLACE TRANFORM

Part A

1.  State the sufficient conditions for the existence of Laplace transform.

2.  Give two examples for which the Laplace transform do not exist. [Ans:]

3.  Give an example of a function such that it has Laplace transform but it is not satisfying the continuity condition. [Ans:]

4.  State & prove first shifting theorem.

5.  If , prove that .

6.  If , prove that .

7.  Define Heaviside’s unit step function; find its Laplace transform.

8.  Define unit impulse (Dirac-delta) function.

9.  State Initial Value & Final Value Theorems.

10.  State convolution theorem.

11.  Define inverse Laplace transform as contour integral.

12.  Find the Laplace transform of the following functions:

[Ans:(i)(ii) (iii) (iv)]

13.  Find the inverse Laplace transform of the following functions:

[Ans(i) (ii) (iii)]

14.  If, find. [Ans:1/2]

15.  If , find [Ans: 0, 1/a]

Part-B

I. State & prove the following results:

(1) Second Shifting Theorem (2) Initial Value Theorem (3) Final Value Theorem

II.Derive the Laplace transform of a periodic function f (t) with the period P.

III.Find the Laplace transform of the unit impulse (Dirac-Delta) function.

IV.Find the Laplace transform of the following functions:

(1) (2) [Ans: (1) (2)]

(3) (4) [Ans: (3) (4)]

(5) (6) [Ans: (5) (6) ]

(7) (8) [Ans:(7)(8)]

V.Given, show that.

VI.Find the Laplace transform of the following functions:

(1) The rectangular-wave function and

(2)The half-sine wave rectifier function and

(3) The square-wave function and

(4)The saw-tooth wave function and

[Ans: (1) (2) (3) (4) ]

VII.Find the inverse Laplace transform of the following functions:

(1) (2) (3) (4) (5) (6)

[Ans:(1)(2)(3) (4)

(5)(6)]

VIII.Verify Initial & Final Value Theorems for the following functions:

1.

2.

IX.Find the inverse Laplace transform of the following functions using convolution theorem:

(1) (2) (3)

(4) (5)

[Ans:1)2)3)4)5)]

X. Solve the following differential equations using Laplace transform:

1. [Ans:]

2. [Ans:]

3. [Ans:]

4. [Ans:]

5. [Ans: ]

6. [Ans: ]

7. [Ans: ]

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