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Convolution Integral
take the inverse transforms of a product of transforms. Fact Let’s work a quick example to see how this can be used. Example 1 Use a convolution integral to find the inverse transform of the following transform. Solution First note that we could use #11 fro...
Periodic Functions and Orthogonal Functions
definition of orthogonal functions out of the way. Definition 1. Two non-zero functions, and , are said to be orthogonal on if, 2. A set of non-zero functions, , is said to be mutually orthogonal on (or just an orthogonal set if we’re be...
Inverse Laplace Transforms
entry in the table has a particular denominator, then the numerators of each will be different, so go up to the numerator and see which one you’ve got. If you need to correct the numerator to get it into the correct form and then take the inverse transform. So, wi...
Nonconstant Coefficient IVP�s
of our solution. So, the transform of our solution, as well as the solution is, I’ll leave it to you to verify that this is in fact a solution if you’d like to. Now, not all nonconstant differential equations need to use (1). So, let’s take a...
Solving IVP�s with Laplace Transforms
an IVP we will need initial conditions at t = 0. While Laplace transforms are particularly useful for nonhomogeneous differential equations which have Heaviside functions in the forcing function we’ll start off with a couple of fairly simple problems to illustrate how the proces...
Step Functions
would be to use the formula to break it up into cosines and sines with arguments of t-5 which will be shifted as we expect. There is an easier way to do this one however. From our table of Laplace transforms we have #16 and using that we can see that if This wi...
IVP�s with Step Functions
the forcing function. This is where Laplace transform really starts to come into its own as a solution method. To work these problems we’ll just need to remember the following two formulas, In other words, we will always need to remember that in order t...
The Definition
can seem scary when we first start looking at them. However, as we will see, they aren’t as bad as they may appear at first. Before we start with the definition of the Laplace transform we need to get another definition out of the way. A function is called piecewise continuous...
Laplace Transforms
will use #24 along with the answer from the previous part. To see this note that if then Therefore, the transform is. [Return to Problems] (e) This final part will again use #30 from the table as well as #35...
Laplace Transforms
Laplace transforms to solve IVP’s. Nonconstant Coefficient IVP’s We will see how Laplace transforms can be used to solve some nonconstant coefficient IVP’s IVP’s with Step Functions Solving IVP’s that contain step functions. This is the section where...
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