# While finding a second solution with wronskian for bessel

• mertymertcan
In summary, the speaker was having difficulty understanding how a negative power can be converted into the right-hand side of an equation, specifically in solving a Bessel differential equation using the Wronskian method. They were looking for a general formula for this technique, and the responder explained that it is called the Frobenius method and provided a detailed explanation using the given equation. The speaker was encouraged to ask any further questions and wished luck with their book.
mertymertcan
Hello guys, I'm new here. i was working on a mathematical methods in physics book and there is a part that i don't understand. so i want to ask if anyone knows... while finding a second solution for bessel diff. eq.(for m=0) the book used wronskian method. in the method there is J^2 bin the denominator. so it becomes J^-2 and the series is made such that

{1 - (x^2)/4 + (x^4)/64 - (x^6)/2304 + ... }^-2 = {1 + (x^2)/2 + (x^6)/32 + ...}

the part that i didnt understand how can someone conver minus 2 power into right hand side:S. please help me, is there a general formula for this?

Hello and welcome to the forum! It's great to have you here. I can understand your confusion about converting the -2 power into the right-hand side of the equation. This is a common technique used in solving differential equations, and it's called the Frobenius method.

The general formula for converting a negative power into the right-hand side is given by the Frobenius method:

y(x) = x^r ∑n=0 an(x-x0)^(n+r)

Where r is the negative power and x0 is the point where the series is expanded. In your case, r = -2 and x0 = 0.

To better understand this, let's take a closer look at your equation:

{1 - (x^2)/4 + (x^4)/64 - (x^6)/2304 + ... }^-2 = {1 + (x^2)/2 + (x^6)/32 + ...}

We can rewrite the left-hand side as:

{1 - (x^2)/4 + (x^4)/64 - (x^6)/2304 + ... }^-2 = (1 - (x^2)/4 + (x^4)/64 - (x^6)/2304 + ...)^-2

Now, using the Frobenius method, we can expand the right-hand side as:

(1 - (x^2)/4 + (x^4)/64 - (x^6)/2304 + ...)^-2 = (1 + (x^2)/2 + (x^6)/32 + ...)^-2

= (1 + (x^2)/2 + (x^6)/32 + ...)^2

= 1 + (2x^2)/2 + (2x^6)/32 + ...

= 1 + x^2 + (x^6)/16 + ...

= 1 + (x^2)/2 + (x^6)/32 + ...

As you can see, the negative power has been converted into the right-hand side using the Frobenius method. I hope this explanation helps you understand the process better. If you have any further questions, please don't hesitate to ask. Good luck with your book!

## What is the Wronskian method and how does it relate to finding a second solution for Bessel's equation?

The Wronskian method is a mathematical technique for determining if two functions are linearly independent. In the context of finding a second solution for Bessel's equation, the Wronskian method is used to verify that the second solution is indeed linearly independent from the first solution.

## Why is it important to find a second solution for Bessel's equation?

For many physical and engineering problems, Bessel's equation is a fundamental equation that describes the behavior of certain systems. However, Bessel's equation has two linearly independent solutions, and both solutions are needed to fully characterize the behavior of the system.

## How is the Wronskian used to find a second solution for Bessel's equation?

The Wronskian method involves taking the determinant of a matrix formed by the two solutions of Bessel's equation. If the determinant is non-zero, then the solutions are linearly independent, and the second solution can be found by using a specific formula involving the first solution and its derivative.

## Are there any limitations to using the Wronskian method for finding a second solution for Bessel's equation?

Yes, the Wronskian method can only be used for finding a second solution for Bessel's equation if the first solution is known. Additionally, the Wronskian method may not work for higher order differential equations or in cases where the solutions of Bessel's equation are non-analytic.

## Can the Wronskian method be applied to other types of equations or systems?

Yes, the Wronskian method is a general technique that can be applied to any set of linearly independent functions or solutions of differential equations. It is commonly used in many areas of physics and engineering to verify the linear independence of solutions and to find additional solutions for various equations.

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