Programmed tutorial: Laplace Transform Example 7

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Find the Laplace transform of
$fractional power$
using the Table of Laplace Transforms.

General Contents

Detailed Contents

Index

Which transform in the table does this look like?

What is the value of n here?

n = 2.

Substitute into #6.

ℒ $Laplace transform of fractional power$

Let’s try an alternate method.
#32 in the table can be applied. Explanation of #32.
We want the result of the integral on the left-hand side to be our t^3/2. What should we try as f(v) under the integral sign?

We know from the Power Rule of Integration that we can integrate t^1/2to get the 3/2 power of t.
Try

for the left-hand side. Do this integral.

We get

We almost have what we need. How can we handle the 3/2 factor?

We could use

Try it.

,
which matches the original problem.

To continue with the right-hand side of #32, we need to evaluate F(s). What does F(s) represent?

It is the Laplace transform of the function we just determined, f(v).

Find F(s) = ℒ {f(t)} = ℒ

ℒ

in the table.

From #5 we get

Use this result to complete the application of #30 to this problem.

ℒ {g(t)} = ℒ $Laplace transform of a fractional power$
This checks with our previous result using #6.

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