Find the inverse Laplace transform of the given function. (2...

Key Concepts

Second Shifting Theorem (Frequency Shifting)

The frequency shifting property (or second shifting theorem) of the Laplace transform involves a shift in the s-domain which corresponds to an exponential multiplication in the time-domain. This property is essential when the Laplace-transformed function contains terms that indicate a shift, enabling one to connect the expression with standard transforms that include exponential factors.

Standard Laplace Transform Pairs

Standard Laplace transform pairs are a set of pre-established relationships between time-domain functions and their Laplace transforms, such as those for sine, cosine, and exponential functions. These pairs allow for quick identification and inversion of Laplace-transformed functions when the given expression can be manipulated to match one of these standard forms.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic expressions in a form that reveals the structure of the denominator. This is particularly useful in finding inverse Laplace transforms, as it helps convert expressions into recognizable standard forms that correspond to known Laplace transform pairs.

Inverse Laplace Transform

The inverse Laplace transform is the process of converting a function from the Laplace domain back into the time domain. It is used to solve differential equations and analyze systems in engineering and physics by mapping complex frequency-domain expressions into time-domain functions using well-known transform pairs and properties.

Transcript

00:01 All right, so our function is 2s plus 1 over s squared minus 2 plus 2 or minus 2 s plus 2.

00:07 So it's not immediately clear what we need to do with this function to find the universal plus transform.

00:13 So our first thing we should do is let's try to simplify it down to a form we might recognize so if we start with the denominator i'm over and then we're gonna get s minus 1 squared plus 1 and if this isn't immediately clear how i did that i'll explain it so if you look at our denominator here 2 s or s squared minus 2 s we can only get that from the factors s minus 1 squared but that would give us if we do that that would give us s squared minus 2 s plus 1 so if we got to be able to plus 2 here we're to add an extra 1 and that is where we get this one on the outside so now we've figured out our denominator.

01:08 If we look at, if we try, if we look at our new denominator, this kind of looks like the laplace transform, which is s minus a over s minus a squared plus b squared, right? where our a term is just one and our b term is also one.

01:39 So now let's try to find a way if we can get a numerator to look like this part of this form.

01:45 So if you do that, we'll take out the two, so we'll have two outside of, we want it to be s minus one.

01:52 So we have two, and then if you multiply the two out, we'll have two s minus two.

01:56 So you make the negative minus two or one, we'll have plus three around the outside.

02:02 So now we have 2 s minus 1 plus 3 over s minus 1 squared plus 1 so now this almost looks like this form over here except we have this 3 here so we're gonna make it that its own fraction so have two s my oh sorry s minus 1 over s minus 1 squared plus 1 plus 3 over over s minus 1 squared plus 1, which i was going to be 1 squared to make that fit with our form...

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