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

`Lap{a · f(t)}=a · Lap{f(t)}`

Example 1

`Lap{7\ sin t}=7\ Lap{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

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

Example 2

`Lap{3t + 6t^2}` `=3 · Lap{t} + 6 · Lap{t^2}`

Property 3. Change of Scale Property

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

Example 3

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

Property 4. Shifting Property (Shift Theorem)

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

Example 4

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

Property 5.

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

See below for a demonstration of Property 5.

Continues below

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`

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

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

Demonstration of Property 5

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

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

Now `F(s)=` `Lap{f(t)}=` `Lap{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:

`Lap{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`

Demonstration of Property 4: Shifting Property

For Example 6 we could have used:

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

Let `g(t) = sin 3t`

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

`=Lap{sin 3t}`

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

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

So

`Lap{e^(2t)sin 3t} =G(s-a)`

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

This is the same result we obtained before for example 6.

Exercises

Find Laplace Transforms of the following.

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

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

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

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

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