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
Example 2 Use Separation of Variables on the following partial differential equation.
In this case we’re looking at the heat equation with no sources and perfectly insulated boundaries.
So, we’ll start off by again assuming that our product solution will have the form,
and because the differential equation itself hasn’t changed here we will get the same result from plugging this in as we did in the previous example so the two ordinary differential equations that we’ll need to solve are,
Now, the point of this example was really to deal with the boundary conditions so let’s plug the product solution into them to get,
Now, just as with the first example if we want to avoid the trivial solution and so we can’t have for every and so we must have,
Here is a summary of what we get by applying separation of variables to this problem.
So, we’ll start off by again assuming that our product solution will have the form,
and because the differential equation itself hasn’t changed here we will get the same result from plugging this in as we did in the previous example so the two ordinary differential equations that we’ll need to solve are,
Now, the point of this example was really to deal with the boundary conditions so let’s plug the product solution into them to get,
Now, just as with the first example if we want to avoid the trivial solution and so we can’t have for every and so we must have,
Here is a summary of what we get by applying separation of variables to this problem.
Next, let’s see what we get if use periodic boundary conditions with the heat equation.
Example 3 Use Separation of Variables on the following partial differential equation.
First note that these boundary conditions really are homogeneous boundary conditions. If we rewrite them as,
it’s a little easier to see.
Now, again we’ve done this partial differential equation so we’ll start off with,
and the two ordinary differential equations that we’ll need to solve are,
Plugging the product solution into the rewritten boundary conditions gives,
and we can see that we’ll only get non-trivial solution if,
So, here is what we get by applying separation of variables to this problem.
it’s a little easier to see.
Now, again we’ve done this partial differential equation so we’ll start off with,
and the two ordinary differential equations that we’ll need to solve are,
Plugging the product solution into the rewritten boundary conditions gives,
and we can see that we’ll only get non-trivial solution if,
So, here is what we get by applying separation of variables to this problem.
Let’s now take a look at what we get by applying separation of variables to the wave equation with fixed boundaries.
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