The solution of the initial-value problem x^2 y^''+x y^'+x^2 y=0 y(0)=1 y^'(0)=0 is cal

The solution of the initial-value problem x^2 y^''+x y^'+x^2...

The solution of the initial-value problem $$x^{2} y^{\prime \prime}+x y^{\prime}+x^{2} y=0 \quad y(0)=1 \quad y^{\prime}(0)=0$$ \begin{equation} \begin{array}{l}{\text { is called a Bessel function of order } 0 \text { . }} \\ {\text { (a) Solve the initial-value problem to find a power series }} \\ {\text { expansion for the Bessel function. }} \\ {\text { (b) Graph several Taylor polynomials until you reach one }} \\ {\text { that looks like a good approximation to the Bessel func- }} \\ {\text { tion on the interval }[-5,5] .}\end{array} \end{equation}

The solution of the initial-value problem
$$x^{2} y^{\prime \prime}+x y^{\prime}+x^{2} y=0 \quad y(0)=1 \quad y^{\prime}(0)=0$$
\begin{equation}
\begin{array}{l}{\text { is called a Bessel function of order } 0 \text { . }} \\ {\text { (a) Solve the initial-value problem to find a power series }} \\ {\text { expansion for the Bessel function. }} \\ {\text { (b) Graph several Taylor polynomials until you reach one }} \\ {\text { that looks like a good approximation to the Bessel func- }} \\ {\text { tion on the interval }[-5,5] .}\end{array}
\end{equation}

Key Concepts

Initial Value Problems

Initial value problems involve differential equations accompanied by specified values of the unknown function and its derivatives at a given point. These problems require determining a unique solution that satisfies both the differential equation and the initial conditions, serving as a cornerstone in the study and application of ordinary differential equations.

Taylor Polynomials

Taylor polynomials provide finite polynomial approximations of functions by truncating their power series. They are used to approximate functions within an interval and are particularly useful for visualizing the behavior of functions, as increasing the degree of the polynomial generally improves the approximation accuracy over a specified range.

Power Series Method

The power series method is a technique for solving differential equations by assuming a solution in the form of an infinite sum. This approach allows one to determine the coefficients of the series by substituting the series into the differential equation, leading to recurrence relations that uniquely define the solution within a radius of convergence.

Bessel Functions

Bessel functions are a family of solutions to Bessel’s differential equation, which arises in many problems with cylindrical symmetry, such as heat conduction, wave propagation, and static potentials. They are characterized by an order parameter and have a series representation that makes them suitable for expressing complex physical phenomena in a compact form.

Transcript

00:01 We have to solve x squared y double prime plus xy prime plus x squared y equals 0 with this power series.

00:09 So we have the derivative of the power series.

00:17 And then we have some initial conditions that y at zero is one, a y prime at zero is zero.

00:25 So we have plugging this in and doing our, you know, the standard process of shifting everything.

00:33 Well, we don't really...

00:36 This winds up giving us an x to the n just to start, and so does this term.

00:42 This gives us an x to the n plus 2, so what we want to do is we want to shift n forward, and so we wind up with a summation from 2 to infinity of c to the n minus 2, c subn minus 2 times x to the n for this term.

01:04 And so let's see here.

01:07 These ones go from 2 to infinity.

01:09 This one goes from 1 in infinity.

01:10 So let's take the first term here, and that's just c1 times x.

01:14 And then the rest of this we can write as this summation here.

01:19 Now, we can also see that if y prime 0 is 0, that means c1 has to be 0.

01:25 Also shows that it basically needs to be 0 anyway from here.

01:30 Now we have, from here, we have c sub n equals minus c sub n minus 2.

01:38 All over n squared.

01:41 So that gives us c2 equals minus c0 over 2 squared.

01:48 C3 is going to be 0 because c1 is.

01:51 C4 is minus c2 all over 4 squared, which is c0 all over 2 squared times 4 squared.

02:00 C5 is going to be 0 because c3 and c1r and c6 is going to be minus c4 all over 6 squared, which is minus c0 over 2 squared times 4 squared times 6 squared.

02:13 And so generalizing this pattern, we see that the even coefficient, c sub 2n, have to toggle the sign, is minus 1 to the n times c0, times divided by, if we look at here, we can pull out a factor of 2 squared.

02:33 And we'll pull out n of those.

02:35 So we get 2 squared to the n.

02:38 And then what we're left with is n factorial squared.

02:44 So that's, let's see here, yeah.

02:52 So i guess if you write this out, you see that it winds up giving us, you know, this sequence here.

03:00 And so simplifying this, we get minus 1 -4 to the n, c -0 over n factorial.

03:07 And now to get their initial conditions, c -0 or y -0 over 0 equals 1, that means c -0 has to be 1.

03:15 So this is 1.

03:18 And so our infinite sequence is, our infinite power series solution is y equals 0 to n from 0 to infinity...

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