The solution to the initial value problem
$$\begin{array}{l}{x y^{\prime \prime}(x)+2 y^{\prime}(x)+x y(x)=0} \\ {y(0)=1, \quad y^{\prime}(0)=0}\end{array}$$
has derivatives of all orders at $x=0$ (although this is far from obvious). Use L' Hopital's rule to compute the Taylor polynomial of degree 2 approximating this solution.

Answer

$$
1-\frac{1}{6} x^{2}
$$

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Calculus 1 / AB

Fundamentals of Differential Equations

Chapter 8

Series Solutions of Differential Equations

Section 1

Introduction: The Taylor Polynomial Approximation

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Video Transcript

So for this problem, we are trying to find thes second degree Taylor family. No meal that approximates the solution to the differential equation. So first of all, let's look at the structure of what the thing is gonna look like. A second degree to LaVoy ovulate was gonna have this basic structure. And luckily for us, we already know most of the pieces. R C in this case is going to be zero. So our initial value conditions are always gonna give already going to give us a lot of the information that we need. So let's see what this looks like when we go ahead and plug those in. Well, if I c equals zero, then we get to plug in. That's why if you're a local one, primacy, zeros of that entire terms gonna zero out, and then we're still missing this f double prime of zero. But other than that, we have all the information we need. So let's figure out with the vegetable prime of zero is going to be so for that, we're gonna need to go back to our initial equation. What we thought this whole differential equation that we're working with we're gonna want to go ahead and solve this for why a double prime of X when X is equal to zero. So our first step is we're gonna subtract off everything that we don't want. And then, ideally, we'd like to for a next divide, everything by X. But because our X is gonna be equal to zero. That's not gonna work. So let's go ahead and just plug in what we know at this step here. Well, that gets us to zero equals zero, which isn't gonna help us out. But what that does mean is that we get to go ahead now and use Lobi tells rule. So let's take the derivative of full size and see where that leaves us. Okay? And now that we've done that, we can go ahead and plug in what we know. So this 1st 1 is gonna zero out because they're X equal to zero y double triumph. X is what we are soling for us. We don't know that yet. This term will zero out. And then why? Zero is one from our initial conditions. So let's clean this up. We get three. Why? Double prime of zero is our ex now gonna be equal to negative one, Which means that y doble crime at zero it's going to be equal to negative one third. Okay? And we are just about done now. This was the only piece of information we were missing. So we've got P of X is now gonna be equal. Teoh. We had our one from before, and then we had after world prime of zero, which we now know is negative 1/3 times X squared over two factorial is going to be, too. If we clean this up, we're going to get that. Are second order Taylor pulling Nobile Could be one minus x squared over six.

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