Programmed tutorial: Inverse Laplace Transform Example 2d

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 “2s” 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.  We could use the “Complete the Square” technique to rewrite it.
See this problem for an example.
Since this denominator is also factorable, let’s try using partial fractions.
Rewrite F(s) with the denominator factored.

Set up the partial fraction equation.

How do we proceed?

We combine fractions in terms of A and B.

Do that.

Distribute the multiplication on the right-hand side.

We get

What do we do next?

We equate coefficients of like powers of s.

Do that.

For  
For

We need to solve these simultaneously.
Substitution looks like an easy way here.
Solve the first equation for B in terms of A.

Substitute this into the second equation.

Solve for A.

Use this to find B.

Use these results to rewrite F(s).

Do these denominators match any in the table?

Yes, they match #2.

What are the values of “a” in each fraction?

a = 5, and a = –3, respectively.

Determine the inverse Laplace transform.

ℒ

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