Question

In Problems 1 and 2 without actually solving the given differential

equation, find the minimum radius of convergence of power series solutions about the ordinary point. About the ordinary point

Short answer

Answer:

About the ordinary point x=0 and x=1, the minimal radius of convergence is 5 and 4, respectively.

Step-by-step solution

Given Information.

The given differential equation is:

Determining the minimum radius of convergence

Our goal is to discover the smallest radius of convergence of the given differential equation's power series solution around the ordinary pointsandFinding the solitary spots where the solution will diverge is the key to answering that issue. We accomplish it by locating the singular points of the differential equation in question. Let's start with the assumption that we have a differential equation of the following form.

Here,

and

Our analysis will be based on equation (1), which is the usual form. The following definition comes to mind.

A point is said to be an ordinary point of the differential equation if underline P(x) and Q(x) are analytic at that point;otherwise,is said to be a singular point of the differential equation.

Using the Talyor series

We may deconstruct the definition as follows: if you have an analytic function at a point, you can expand it around that point using the Talyor Series, which is calculated using the function's derivatives at that location. You won't be able to identify the function's derivatives at that moment if the function contains singular points, and it will blow up. Investigating (2), you can determine thatandare not analytic at the points, causing the denominator to equal zero after removing common factors. We now convert the provided differential equation to standard form.

It gives,

and

Now put the denominator equal to 0 and look for numbers that satisfy this criterion.

As a result, atand, andare not analytic, implying that the differential equation contains singular points atand. We've arrived at the unique spots. It's now up to us to identify the smallest radius of convergence between xand.

If, then:

As a result, the convergence radius aroundhas a minimum radius of 5.

If, then:

and

As a result, the minimum radius of convergence aroundis.

About the ordinary pointand, the minimal radius of convergence is 5 and 4, respectively.