In this section we need to take a look at the equation of a line in R 3 R 3 . As we saw in the previous section the equation y = m x + b y = m x + b does not describe a line in R 3 R 3 , instead it describes a plane. This doesn’t mean however that we can’t write down an equation for a line in 3-D space. We’re just going to need a new way of writing down the equation of a curve. So, before we get into the equations of lines we first need to briefly look at vector functions. We’re going to take a more in depth look at vector functions later. At this point all that we need to worry about is notational issues and how they can be used to give the equation of a curve. The best way to get an idea of what a vector function is and what its graph looks like is to look at an example. So, consider the following vector function. → r ( t ) = ⟨ t , 1 ⟩ r → ( t ) = ⟨ t , 1 ⟩ A vector function is a function that takes one or more variables, one in this case, and returns a...
Example 3 Solve the following IVP.
Let’s take the Laplace transform of everything and note that in the third term we are shifting 4.
Now solve for .
So, we have three functions that we’ll need to partial fraction for this problem. I’ll leave it to you to check the details.
Okay, we can now get the solution to the differential equation. Starting with the transform we get,
where , and are given above.
Now solve for .
So, we have three functions that we’ll need to partial fraction for this problem. I’ll leave it to you to check the details.
Okay, we can now get the solution to the differential equation. Starting with the transform we get,
where , and are given above.
Let’s work one more example.
Example 4 Solve the following IVP.where,
The first step is to get written in terms of Heaviside functions so that we can take the transform.
Now, while this is written in terms of Heaviside functions it is not yet in proper form for us to take the transform. Remember that each function must be shifted by a proper amount. So, getting things set up for the proper shifts gives us,
So, for the first Heaviside it looks like is the function that is being shifted and for the second Heaviside it looks like is being shifted.
Now take the Laplace transform of everything and plug in the initial conditions.
Solve for .
Now, in the solving process we simplified things into as few terms as possible. Even doing this, it looks like we’ll still need to do three partial fractions.
We’ll leave the details of the partial fractioning to you to verify. The partial fraction form and inverse transform of each of these are.
Putting this all back together is going to be a little messy. First rewrite the transform a little to make the inverse transform process possible.
Now, taking the inverse transform of all the pieces gives us the final solution to the IVP.
where , , and are defined above.
Now, while this is written in terms of Heaviside functions it is not yet in proper form for us to take the transform. Remember that each function must be shifted by a proper amount. So, getting things set up for the proper shifts gives us,
So, for the first Heaviside it looks like is the function that is being shifted and for the second Heaviside it looks like is being shifted.
Now take the Laplace transform of everything and plug in the initial conditions.
Solve for .
Now, in the solving process we simplified things into as few terms as possible. Even doing this, it looks like we’ll still need to do three partial fractions.
We’ll leave the details of the partial fractioning to you to verify. The partial fraction form and inverse transform of each of these are.
Putting this all back together is going to be a little messy. First rewrite the transform a little to make the inverse transform process possible.
Now, taking the inverse transform of all the pieces gives us the final solution to the IVP.
where , , and are defined above.
So, the answer to this example is a little messy to write down, but overall the work here wasn’t too terribly bad.
Before proceeding with the next section let’s see how we would have had to solve this IVP if we hadn’t had Laplace transforms. To solve this IVP we would have had to solve three separate IVP’s. One for each portion of . Here is a list of the IVP’s that we would have had to solve.
- The solution to this IVP, with some work, can be made to look like,
- where, is the solution to the first IVP. The solution to this IVP, with some work, can be made to look like,
- where, is the solution to the second IVP. The solution to this IVP, with some work, can be made to look like,
There is a considerable amount of work required to solve all three of these and in each of these the forcing function is not that complicated. Using Laplace transforms saved us a fair amount of work.
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