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)} = alap{f(t)}

 

Example

lap{7 sin t} = 7lap{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)} = alap{f(t)} + blap{g(t)}


Example

lap{3t + 6t2 } = 3 lap{t} + 6lap{t2}


Property 3. Change of Scale Property

If lap{f(t)} = F(s) then MATH


Example

change of scale


Property 4. Shifting Property (Shift Theorem)

lap{eatf(t)} = F(sa)


Example

lap{e3tf(t)} = F(s − 3)


Property 5.

MATH


Property 6.

The Laplace transforms of the real (or imaginary) part of a complex function is equal to the real (or imaginary) part of the transform of the complex function.

Let Re denote the real part of a complex function C(t) and Im denote the imaginary part of C(t), then

lap{Re[C(t)]} = Re lap{C(t)}

and

lap{Im[C(t)]} = Im lap{C(t)}

If you need some background, go to Complex Numbers.


EXAMPLES

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) = 4t2


Answer



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


Answer



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


Answer


DEMONSTRATION of PROPERTY 5:

lap{t f(t)}

For example (c), we could have also used Property 5:

MATH

with f(t) = cos 7t.

Now MATH

So

MATH

So MATH


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.



(d) f(t) = e2t sin 3t


Answer


DEMONSTRATION OF No 4: SHIFTING PROPERTY

For example (d) we could have used:

lap{eatg(t)} = G(sa)

Let g(t) = sin 3t

MATH

So MATH


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


(e) f(t) = t4e-jt


Answer



(f) f(t) = te-t cos 4t


Answer



(g) f(t) = t2 sin 5t


Answer



(h) f(t) = t3 cos t = t2(t cos t)


Answer



(i) f(t) = cos23t, given that MATH


Answer




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