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University of North Texas

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Problem 8 Easy Difficulty

SturmLiouville Form. A second-order equation is said to be in SturmLiouville form if it is expressed as
$$\left[p(t) y^{\prime}(t)\right]^{\prime}+q(t) y(t)=0$$
Show that the substitutions $x_{1}=y, x_{2}=p y^{\prime}$ result inthe normal form
$$\begin{aligned} x_{1}^{\prime} &=x_{2} / p \\ x_{2}^{\prime} &=-q x_{1} \end{aligned}$$

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Video Transcript

So they want us to make that substitution, um, to result in the normal form that they had a list of there. So let's go ahead and do that. So what they first say is, OK, well, let's make this here X two, and then why is this gonna be X one? And if we were to plug those that now so there's going to be next to Well, that's gonna be prime Plus and I'm just gonna write that is Q and then we have X one here and then that's equal to zero. So and for solving for X two, we subtract that over. So it be x two Prime is able to negative q x one, which is the first one they wanted us to show. And then the second one, What we need for a wood, um, X one is well, remember, over here we said x one was equal toe. Why want to get the derivative of that? We could just take the derivative on each side because we know the derivative of wise just why prime And now in order for us to get it into, like that form that we had before or at least what they suggested we would come over here to what? Our substitution for X two waas. So next two is equal to p of tea. And actually, I should just write those p so p times why prime? And then we could just solve for y prime. So I'll divide p over. So that says why Prime is equal to X over to p. And then we can just plug that in down here, and that gives x one prime is equal to x two over p. And so you could say we end up with that solution that, uh, that they wanted us to do. The next thing is, they tell us that why zero is equal to a and why prime of zero is equal to be. And they want us to find what is x one of zero as well as what is x two of zero? Well, from those substitution is we made, um, where was it? S o X one is just going to be. Why one or not? Why? Why? Just why of zero, which they told us was just a so that was straightforward enough. And then why to Well, if we come back up here, This is will speak p times. Why Prime? So this is going to be p of zero times. Why? Prime of zero. And I don't really think there's a way for us to find what P zero is. So we can just go ahead and leave it like that, at least for right now. Um, but then we can plug in UAE, prime of zero, which should be be. So this is going to be p of zero times be mhm. And uh huh. So, for this would just be X one of zero is a and X two of zero is p of zero times be. Yeah, I kind of unfortunate. There is a really nice way to find that. But we would just have to kind of know what p of X is