Programmed tutorial: Inverse Laplace Transform Example 2b

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Find the inverse Laplace transform of

How do we go about finding entries in the Table of Laplace Transforms?

For each term, we compare its denominator with those in the table.

Which entry in the table matches the denominator?

None seem to directly match.

What term in the denominator seems to be missing in the table?

The “10s” term is missing. The entries in the table don’t seem to contain a term that is linear in “s”.

Our denominator is a quadratic. Can we use the “Complete the Square” technique to rewrite it?

Yes.

Do that.

By adding and subtracting the same thing, we get

Rewrite F(s) with this new expression in the denominator.

Does the denominator match any of those in the table?

#19 and #20.

What are our values of “a” and “b”?

Does the numerator match those in #19 or #20?

No.

19Our numerator may be a linear combination of those from #19 and #20. Try regrouping to get a multiple of

We get

Rewrite F(s) using this result for the numerator.

Rewrite it as the sum of two fractions.

Do we now have #19 and #20?

Almost. The first term needs a

factor.

How can we introduce this?

By multiplying by

Rewrite accordingly.

We get

Find the inverse Laplace transform for this expression.

We get
ℒ Inverse Laplace Transform, non-factorable quadratic in denominator

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