chapter 9(laplace transform)
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Engineering Mathematics 1TRANSCRIPT
BMM 104: ENGINEERING MATHEMATICS I Page 1 of 8
CHAPTER 9: THE LAPLACE TRANSFORM
Laplace Transformation and its Inverse
Definition:
Let be defined for and let s denote an arbitrary real variable. The Laplace transform of , designated by either or , is
for all values of s for which the improper integral converges. Convergence occurs when the limit
exists. If the limit does not exist, the improper integral does not exist, the improper integral diverges and has no Laplace transform. When evaluating the integral, the variable s is treated as a constant because the integration is with respect to t.
On the other hand, we may write with is called inverse Laplace transformation operator and is called the inverse Laplace transformation for .
Example:
Find the Laplace transform by definition.
(a) (b)
Properties ( Linearity of Laplace Operator L and its inverse )
Suppose and are arbitrary constants, then
(i)
(ii)
Example:
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Find the Laplace transform for the following functions.
(a)
(b)
(c)
Example:
(a) Find .
(b) Find .
(c) Find .
(d) Find .
(e) Find .
Theorem:
Suppose is a continuous function for that has Laplace transform . If and have Laplace transformation, then
*
*
NOTE:
The above Theorem is applied in solving initial value problem.
We apply,
and
where .
Example:
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By taking Laplace transformation on both of the following differential equations, find
(a)(b)(c)
First Shift Theorem
If then where a is a real constant.
Example:
Determine the following.
(a) (b)
Multiplication by t Theorem
If then .
Example:
Obtain the following.
(a) (b) (c) (d)
Solving Linear Initial-Value Problems with Constant Coefficients
Laplace transform for derivatives of a function contain terms that need the values of the function and its derivative at t = 0. By having these (initial) conditions, the approach
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using Laplace transformation become very suitable to solve initial value problem that involving constant coefficients.
Example:
(a) Solve .(b) Solve .(c) Solve .
TABLE OF LAPLACE TRANSFORMS
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1 1
2 t
3 ,
4
5
6 0s
7
8
9
10
11
12
13
14 atsinht as
15
16 with 171819 ,
20 atfLe as
21 tfLe as
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22 is periodic with period
23
24
PROBLEM SET: CHAPTER 9
1. Evaluate the following Laplace transform.
(a)
(b)(c)(d)(e)
2. Solve by using First Shift Theorem.
(a)
(b)
(c)
3. Solve by using Multiplication by t Theorem.
(a) (b)
(c)
4. Find the inverse of the following Laplace transform.
(a)
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(b)
(c)
(d)
(e)
5. Solve the following initial value problem.
(a) ;(b) ;(c) ;(d) ;(e) ;
ANSWERS FOR PROBLEM SET: CHAPTER 9
1. (a)
(b)
(c)
(d)
(e)
2. (a)
(b)
(c)
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3. (a)
(b)
(c)
4. (a)
(b)
(c)
(d)
(e)
5. (a)
(b) OR
(c)
(d)
(e)