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 1 Find a general solution to the following differential equation.
First, since the formula for variation of parameters requires a coefficient of a one in front of the second derivative let’s take care of that before we forget. The differential equation that we’ll actually be solving is
We’ll leave it to you to verify that the complementary solution for this differential equation is
So, we have
The Wronskian of these two functions is
The particular solution is then,
The general solution is,
We’ll leave it to you to verify that the complementary solution for this differential equation is
So, we have
The Wronskian of these two functions is
The particular solution is then,
The general solution is,
Example 2 Find a general solution to the following differential equation.
We first need the complementary solution for this differential equation. We’ll leave it to you to verify that the complementary solution is,
So, we have
The Wronskian of these two functions is
The particular solution is then,
The general solution is,
So, we have
The Wronskian of these two functions is
The particular solution is then,
The general solution is,
This method can also be used on non-constant coefficient differential equations, provided we know a fundamental set of solutions for the associated homogeneous differential equation.
Example 3 Find the general solution togiven that
form a fundamental set of solutions for the homogeneous differential equation.
form a fundamental set of solutions for the homogeneous differential equation.
As with the first example, we first need to divide out by a .
The Wronskian for the fundamental set of solutions is
The particular solution is.
The general solution for this differential equation is.
The Wronskian for the fundamental set of solutions is
The particular solution is.
The general solution for this differential equation is.
We need to address one more topic about the solution to the previous example. The solution can be simplified down somewhat if we do the following.
Now, since is an unknown constant subtracting 2 from it won’t change that fact. So we can just write the as and be done with it. Here is a simplified version of the solution for this example.
This isn’t always possible to do, but when it is you can simplify future work.
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