Programmed tutorial: Laplace Transform Example 8

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Find the Laplace transform of
$fractional power$
using #24 in the Table of Laplace Transforms. Explanation of #24.

How can we use #24?

We need to rewrite the function g(t) as depending on a constant times a variable, or as f(ct).

What is the possible function for f here?

The 3/2 power function.

Rewrite g(t) in this way, if possible.

Try
$scaled fractional power$

Solve for a value of c.

We have

, or

Use this in g(t).

We get
$scaled fractional power$

Next determine F(s), the Laplace transform of the 3/2 power function.

From #6 in the table, we get
Laplace transform of the 3/2 power

Now evaluate #24 with these values.

Simplify.

We get

Check this result by using #6 in the table.

With

, we have n = 2. #6 becomes
Laplace transform of the scaled 3/2 power

, which checks.

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