The solution to the initial value problem x y^''(x)+2 y^'(x)+x y(x)=0 y(0)=1, y^'(0)

The solution to the initial value problem x y^''(x)+2...

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.

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.

Key Concepts

Taylor Series

A Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor polynomial of a certain degree is a finite truncation of this series that approximates the function around that point, with higher-degree polynomials generally offering more accurate approximations.

Initial Value Problem

An initial value problem (IVP) is a differential equation paired with initial conditions that specify the value of the solution and possibly some of its derivatives at a particular point. These conditions ensure a unique solution exists in a neighborhood around that point.

L'Hopital's Rule

L'Hopital's rule is a technique used to evaluate limits that result in indeterminate forms such as 0/0 or ?/?. By differentiating the numerator and denominator and then taking the limit of the resulting expression, one can often resolve the indeterminacy and determine the desired limit.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations involving functions of a single variable and their derivatives. They are fundamental in modeling dynamic systems, and techniques such as series expansions and analytic methods are applied to find solutions, particularly near points where the behavior of the solution is well understood.

Transcript

00:01 So for this problem, we are trying to find the second degree taylor polynomial that approximates the solution to the differential equation.

00:09 So first of all, let's look at the structure of what this thing is going to look like.

00:13 A second degree tailor polynomial is going to have this basic structure, and luckily for us, we already know most of the pieces.

00:21 Our c in this case is going to be zero, so our initial value conditions are already going to give us a lot of the information that we need.

00:30 So let's see what this looks like when we go ahead and plug those in.

00:33 Well, if our c equals zero, then we get to plug in that it's y of zero equals 1.

00:38 If prime of c is zero, so that entire term's going to zero out.

00:42 And then we are still missing this f double prime of zero.

00:47 But other than that, we have all the information we need.

00:50 So let's figure out what that f double prime of zero is going to be.

00:56 So for that, we're going to need to go back to our initial equation, where we've got this whole differential equation that we'll we're working with.

01:10 And we're going to want to go ahead and solve this for y double prime of x when x is equal to 0.

01:16 So our first step is we're going to subtract off everything that we don't want.

01:29 And then ideally we'd like to for our next step, divide everything by x.

01:33 But because our x is going to be equal to zero, that's not going to work.

01:37 So let's go ahead and just plug in what we know at this step here...

Please give Ace some feedback