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 vector. Note as we
There really isn’t all that much to this section. All we’re going to do here is work a quick example using Laplace transforms for a 3rd order differential equation so we can say that we worked at least one problem for a differential equation whose order was larger than 2.
Okay, there is the one Laplace transform example with a differential equation with order greater than 2. As you can see the work in identical except for the fact that the partial fraction work (which we didn’t show here) is liable to be messier and more complicated.
Everything that we know from the Laplace Transforms chapter is still valid. The only new bit that we’ll need here is the Laplace transform of the third derivative. We can get this from the general formula that we gave when we first started looking at solving IVP’s with Laplace transforms. Here is that formula,
Here’s the example for this section.
Example 1 Solve the following IVP.
As always, we first need to make sure the function multiplied by the Heaviside function has been properly shifted.
It has been properly shifted and we can see that we’re shifting . All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for . Doing all of this gives,
Now we need to partial fraction and inverse transform and . We’ll leave it to you to verify the details.
Okay, we can now get the solution to the differential equation. Starting with the transform we get,
where and are the functions shown above.
It has been properly shifted and we can see that we’re shifting . All we need to do now is take the Laplace transform of everything, plug in the initial conditions and solve for . Doing all of this gives,
Now we need to partial fraction and inverse transform and . We’ll leave it to you to verify the details.
Okay, we can now get the solution to the differential equation. Starting with the transform we get,
where and are the functions shown above.
Okay, there is the one Laplace transform example with a differential equation with order greater than 2. As you can see the work in identical except for the fact that the partial fraction work (which we didn’t show here) is liable to be messier and more complicated.
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