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...
Now that we’ve got some of the basics out of the way for systems of differential equations it’s time to start thinking about how to solve a system of differential equations. We will start with the homogeneous system written in matrix form,
where, is an matrix and is a vector whose components are the unknown functions in the system.
Now, if we start with then the system reduces to a fairly simple linear (or separable) first order differential equation.
and this has the following solution,
So, let’s use this as a guide and for a general let’s see if
will be a solution. Note that the only real difference here is that we let the constant in front of the exponential be a vector. All we need to do then is plug this into the differential equation and see what we get. First notice that the derivative is,
So, upon plugging the guess into the differential equation we get,
Now, since we know that exponentials are not zero we can drop that portion and we then see that in order for to be a solution to then we must have
Or, in order for to be a solution to , and must be an eigenvalue and eigenvector for the matrix .
Therefore, in order to solve we first find the eigenvalues and eigenvectors of the matrix and then we can form solutions using . There are going to be three cases that we’ll need to look at. The cases are real, distinct eigenvalues, complex eigenvalues and repeated eigenvalues.
None of this tells us how to completely solve a system of differential equations. We’ll need the following couple of facts to do this.
Fact
- If and are two solutions to a homogeneous system, , thenis also a solution to the system.
- Suppose that is an x matrix and suppose that , , …, are solutions to a homogeneous system, . Define,In other words, is a matrix whose ith column is the solution. Now define,
We call the Wronskian. If then the solutions form a fundamental set of solutions and the general solution to the system is,
Note that if we have a fundamental set of solutions then the solutions are also going to be linearly independent. Likewise, if we have a set of linearly independent solutions then they will also be a fundamental set of solutions since the Wronskian will not be zero.
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