# What did I do wrong in this Bessel equation?

• yungman
In summary, the person is trying to convert a given equation using the substitution y=x^{-\frac{1}{2}}w to a Modified Bessel Equation. They have made some arithmetic mistakes but have eventually arrived at the correct equation with (n+\frac{1}{2})^{2} as the order. They attribute their mistake to old age but are determined to overcome it.
yungman
I need to convert$$x^{2}y''+2xy'+[kx^{2}-n(n+1)]y=0$$ using $$y=x^{-\frac{1}{2}}w$$ to a normal Modified Bessel Equation and I cannot get to that. I check many times and I must be having a blind spot!

This is my work:

$$y=x^{-\frac{1}{2}} w \Rightarrow y'=-\frac{1}{2}x^{-\frac{3}{2}}w+x^{-\frac{1}{2}}w' \Rightarrow y''=\frac{3}{4}x^{-\frac{5}{2}}w-\frac{1}{2}x^{-\frac{3}{2}}w'+x^{-\frac{1}{2}}w''-\frac{1}{2}x^{-\frac{3}{2}}w'$$

$$x^{2}y''+2xy'+(kx^{2}-n(n+1))y=0 \Rightarrow$$

$$[\frac{3}{4}x^{-\frac{1}{2}}w-\frac{1}{2}x^{\frac{1}{2}}w'+x^{\frac{3}{2}}w''-\frac{1}{2}x^{\frac{1}{2}}w']+[2x^{\frac{1}{2}}w'-x^{-\frac{1}{2}}w]+[kx^{\frac{3}{2}}w-n(n+1)x^{-\frac{1}{2}})w]=0$$

Grouping w'', w' and w terms:

$$x^{\frac{3}{2}}w''+(-x^{\frac{1}{2}}+2x^{\frac{1}{2}})w'+(\frac{3}{4}x^{-\frac{1}{2}}-x^{-\frac{1}{2}}+kx^{\frac{3}{2}}-n(n+1)x^{-\frac{1}{2}})w=0$$

Multiply by $$x^{\frac{1}{2}}$$ This give:

$$x^{2}w''+xw'+(\frac{3}{4}-1+kx^{2}-n(n+1))w=0$$

I can't get the standard Modified Bessel equation because the 3/4-1 in the w term cannot be cancel out.
Please tell me what I did wrong.

Thanks
Alan

Last edited:
Your arithmetic is correct (except that in the first line you have an exponent $$-\frac23$$ that should be $$-\frac32$$, but later steps are correct).

Is there any reason why the $$n$$ (order) in the equation you end up with is required to be the same you began with? I notice that your problematic constant term is $$-n^2 - n - \frac14$$, which factors nicely.

ystael said:
Your arithmetic is correct (except that in the first line you have an exponent $$-\frac23$$ that should be $$-\frac32$$, but later steps are correct).

Is there any reason why the $$n$$ (order) in the equation you end up with is required to be the same you began with? I notice that your problematic constant term is $$-n^2 - n - \frac14$$, which factors nicely.

You are right, I was blind! The final equation is with $$(n+\frac{1}{2})^{2}$$!

It must be old age! I read the final equation wrong and I just kept concentrate on what I did and never look at the final equation! Wasted almost an hour on this! The only excuse I can come up is I am 56! Too old! BUT I am not old enough to conceed to you guys yet!

Thanks a million.

Alan

Last edited:

## 1. What is a Bessel equation?

A Bessel equation is a type of differential equation that is used to describe oscillatory or wave-like behavior in various physical systems, such as vibrating strings, heat transfer, and fluid dynamics.

## 2. Why is it important to understand Bessel equations?

Bessel equations have many important applications in physics, engineering, and mathematics. They are used to model a wide range of physical phenomena and can provide valuable insights into the behavior of these systems.

## 3. What are the common mistakes made when solving a Bessel equation?

Some common mistakes when solving a Bessel equation include using incorrect boundary conditions, making errors in the algebraic manipulation of the equation, and not considering the various special cases and boundary conditions that may arise.

## 4. How can I avoid making mistakes when working with Bessel equations?

To avoid mistakes when working with Bessel equations, it is important to carefully read and understand the problem, use the correct boundary conditions, and double-check all algebraic manipulations. It can also be helpful to work through examples and practice problems to gain familiarity with the equations and their solutions.

## 5. Are there any tips for effectively solving Bessel equations?

One tip for solving Bessel equations is to use known solutions or properties of Bessel functions to simplify the equation. It can also be helpful to break the equation into smaller, more manageable parts and to use numerical methods when necessary. Additionally, it is important to regularly check your work and verify that the solution satisfies the original equation and boundary conditions.

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