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Unit VI

Laplace Transforms

Definition:- If f(t) is a real valued function for all , then the laplace transform of f(t), denoted

by L{f(t)} and is defined as

Where s is parameter, real or complex.

Note: where a and b are constants.

Laplace transform of standard functions:-

1. Prove that, , where a is constant.

Proof: By definition

If a=1, then

2. Prove that

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Proof: By definition

3. Prove that

Proof:

4. Prove that

Proof:

5. Prove that

Proof: By definition

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6. Prove that

Proof: By definition

7. Prove that

Proof: By definition

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where n is a positive integer

8. Prove that, where n is constant

Proof: By definition

put st=x , s dt= dx, dt=dx/s

when t=0, x=0 and when t=, x=

therefore L{

Definition: replacing n by n+1

Examples:

1. Find L{f(t)} where f(t)=

Solution: By definition,

L{f(t)} =

=

=

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=

=

2. Find the Laplace transform of f(t) =

Solution : By definition,

L{f(t)} =

=

=

= + 5

= - {4-0}- {-1}- {0-}

= - - {-1}+

= - {-1}

3. Find the Laplace transform of f(t)=

Solution: By definition,

L{f(t)} =

=

=

=

=

=

4. Find the Laplace transform of f(t)=

Solution: L{f(t)} =

5. Find the Laplace transform of f(t)=

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First shifting property: If L{f(t)} = f(s) then, L{f(t)} = f(s-a)

Proof: By definition,

L{f(t)} =

=

= f(s-a)

Standard results:

1. L{}= because, L{}=

2. L{}= because, L{}=

3. L{}= because, L{}=

4. L{}= because, L{}=

5. L{}= because, L{}=

Examples:

1. Find L{}

We know that,

Therefore L{} =

=

2. Find L{t}

We know that, = (put =2t)

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t} = L{ }

= [L{1}+L{}]

= { +}

3. Find L{t}

We know that sin3 = 3sin - 4 or (put

L{t} = L{ }

= {sin2t} - {sin6t}

= {} - {}

4. Find L{t cos2t cos3t}

We know that cosA cosB = (put

L{t cos2t cos3t} =

=

=

=

=

5. Find L{ +4 -2sin3t +3cos3t}

L{ +4 -2sin3t +3cos3t} = +4() -2( +3()

6. Find L{ }

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L{ } = L{

= L{ 1-sin2t}

= -

7. Find L{sin2t cos3t}

We know that, cosA sinB = {sin(A+B)-sin(A-B)}

L{sin2t cos3t} = L{sin5t-sint}

= { -}

8. Find L{ }

We know that cos3 = 4- 3cos (put

L{ } = L{ }

=

= }

9. Find L{ sin5t cos2t }

10. L { }

11. L { + }

12. L { }

13. Find L{ }

14. Find L{ }

L{ } = L{ (

= L{

=

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15. Find L{ }

L{ } = 2L {{ sin5t}

= 2 {} - 3 {}

=

16. Find L{ }

L{ } = L{ } because, =

= L{ }

= L{ }

17. Find L{ }

We know that, sinA cosB = {sin(A+B)+sin(A-B)}

L{ } = L{ }

= L{ }

= +}

18. Find L{ }

L{ } =}

19. L{ }

20. L{sinhat sinat}

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Properties of Laplace transforms:

Property 1: If L{f(t)}= f(s) then L{

Proof: By definition

Differentiating both sides with respect to s

f(s)] = f(t) dt

According to Leibnitz rule for differentiation under integral sign

f(s)] = f(t) dt

f(s)] = f(t) dt

f(s)] = - f(t) dt

(-1)f(s)] = L{t f(t)}

Similarly

Property 2: If then prove that

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Proof: By definition

Integrating both sides between s to with respect to s

[changing the order of integration]

Examples:

(1) Find

We know that

(2) Find

We know that

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(3) Find L{t2 sin at}

We know that

Therefore

(4) Find

5. Find

We know that

Therefore

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6. Find

7. Find

We know that

Therefore

8. Find

We know that

Therefore

9. Find

We know that

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10. Find

L{

11. Find

We know that

12. Find

13. Find

14. Show that

Put s=2

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15. P.T.

We have,

Put s=3,

16. Evaluate using Laplace transform

We have,

Put s=0

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Property 3: If then

Proof: Let and hence and

17. Find

Let

18. Find

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Let

L[