Chapter Contents

• Laplace Transforms
• 1a. The Unit Step Function - Definition
• Oliver Heaviside
• 1b. The Unit Step Function - Products
• 2. Laplace Transform Definition
• Table of Laplace Transformations
• 3. Properties of Laplace Transform
• 4. Transform of Unit Step Functions
• 5. Transform of Periodic Functions
• 6. Transforms of Integrals
• 7. Inverse of the Laplace Transform
• 8. Using Inverse Laplace to Solve DEs
• 9. Integro-Differential Equations and Systems of DEs
• 10. Applications of Laplace Transform
• Laplace Transforms Problem Solver

• Comments, Questions?

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3. Some Properties of Laplace Transforms

We saw some of the following properties in the Table of Laplace Transforms.

Property 1. Constant Multiple

If a is a constant and f(t) is a function of t, then

`mathcal(L){a · f(t)}=a · mathcal(L){f(t)}`

Example 1

`mathcal(L){7\ sin\ t}=7\ mathcal(L){sin\ t}`

[This is not surprising, since the Laplace Transform is an integral and the same property applies for integrals.]

Property 2. Linearity Property

If a and b are constants while f(t) and g(t) are functions of t, then

`mathcal(L){a · f(t) + b · g(t)}` `= a · mathcal(L){f(t)} + b · mathcal(L){g(t)}`

Example 2

`mathcal(L){3t + 6t^2}` `=3 · mathcal(L){t} + 6 · mathcal(L){t^2}`

Property 3. Change of Scale Property

If `mathcal(L){f(t)}=F(s)` then `ccL{f(at)}=1/aF(s/a)`

Example 3

`ccL{f(5t)}=1/5F(s/5)`

Property 4. Shifting Property (Shift Theorem)

`ccL {e^(at)f(t)} = F(s-a)`

Example 4

`ccL {e^(3t)f(t)} = F(s-3)`

Property 5.

`ccL{tf(t)}=-F^'(s)=-d/(ds)F(s)`

See below for a demonstration of Property 5.

Example 5

Obtain the Laplace transforms of the following functions, using the Table of Laplace Transforms and the properties given above.

(We can, of course, use Scientific Notebook to find each of these. Sometimes it needs some more steps to get it in the same form as the Table).

(a) `f(t) = 4t^2`

Answer

(b) `v(t) = 5\ sin\ 4t`

Answer

(c) `g(t) = t\ cos\ 7t`

Answer

Demonstration of Property 5

Example (c) is of the form `ccL{tf(t)}`.

We could have also used Property 5, `ccL{tf(t)}=-F^'(s)=-(d/(ds)F(s))`, with `f(t) = cos\ 7t`.

Now `F(s)=ccL{f(t)}` `=ccL{cos\ 7t}` `=s/(s^2+7^2)`

So

`d/(ds)F(s)`

`=d/(ds)s/(s^2+7^2)`

`=-(s^2-7^2)/(s^2+7^2)^2`

`=-(s^2-49)/(s^2+49)^2`

Then we have:

`ccL{t\ cos\ 7t}`

`=-(-(s^2-49)/(s^2+49)^2)`

`=(s^2-49)/(s^2+49)^2`

This is the same result that we obtained using the formula.

For a reminder on derivatives of a fraction, see Derivatives of Products and Quotients.

Example 6

Find the Laplace Transform of `f(t)=e^(2t)sin\ 3t`

Answer

Demonstration of Property 4: Shifting Property

For Example 6 we could have used:

`ccL {e^(at)g(t)} = G(s-a)`

Let `g(t) = sin\ 3t`

`G(s)=ccL{g(t)}`

`=ccL{sin\ 3t}`

`=3/{s^2+3^2)`

`=3/(s^2+9)`

So

`ccL{e^(2t)sin\ 3t}` `=G(s-a)` `=3/((s-2)^2+9)`

This is the same result we obtained before for example (d).

Exercises

Find Laplace Transforms of the following.

1. `f(t)=t^4e^(-jt)`

Answer

2. `f(t) = te^(-t)\ cos\ 4t`

Answer

3. `f(t) = t^2sin\ 5t`

Answer

4. `f(t) = t^3cos\ t = t^2(t\ cos\ t)`

Answer

5. `f(t)=cos^2 3t` given that `ccL{cos^2t}=(s^2+2)/(s(s^2+4))`

Answer