Showing that a bessel function satisfies a particular equation

In summary, The conversation discusses solving the equation y'' + ((1+2n)/x)y' + y = 0 and verifying that it is satisfied by the function x-nJn(x). The conversation includes an attempt at a solution using Bessel functions and a suggestion to use the product rule instead. The final response advises to solve the equation without using special properties of the Bessel functions.
  • #1
1Kris
22
0
Hi, I'm stuck on this question from a calculus book;
Show that y'' + ((1+2n)/x)y' + y = 0 is satisfied by x-nJn(x)

Is it correct that when I differentiate that, I get these:
y= x-nJn(x)
y'=-x-nJn+1(x)
y''=nx-n-1Jn+1(x) -
x-n(dJn+1(x)/dx)?

The Attempt at a Solution


Equation in question becomes:
x-n(2(n/x)Jn+1 - Jn - ((1+2n)/x)Jn+1 + Jn)

= x-n(-x-1Jn+1)
but this isn't 0.

Sorry if I'm repeating myself here but I could just do with some kind of a pointer.
Thanks, Kris
 
Physics news on Phys.org
  • #2
Shouldn't y' = -nx-n-1Jn(x) + x-nJ'n(x) ? You have to use the product rule.
 
  • #3
I think the OP is using some property of the Bessel functions to get that expression for y'. I could very well be wrong though; I don't remember much about the Bessel functions.
 
  • #4
Just crank it out, without resorting to any special properties and relations of the Bessel functions. Use the product rule to get y' and y'', plug them into the differential equation, and simplify.
 

FAQ: Showing that a bessel function satisfies a particular equation

What is a Bessel function and why is it important?

A Bessel function is a special type of mathematical function that is widely used in physics and engineering to describe various physical phenomena, such as wave propagation and heat conduction. It was first introduced by the mathematician Friedrich Bessel in the early 19th century and has since been extensively studied and applied in many fields of science and engineering.

What is the equation that a Bessel function satisfies and why is it significant?

A Bessel function satisfies a second-order differential equation known as the Bessel equation. This equation is significant because it arises in many physical problems involving cylindrical or spherical symmetry, such as the motion of a pendulum or the propagation of electromagnetic waves. Solving this equation allows us to understand and predict the behavior of these physical systems.

How do you show that a Bessel function satisfies a given equation?

To show that a Bessel function satisfies a particular equation, we first need to substitute the Bessel function into the equation and then use its properties and known identities to simplify the equation. This will eventually lead to an expression that is equivalent to the Bessel equation, thus proving that the Bessel function satisfies the given equation.

Are there any other ways to prove that a Bessel function satisfies an equation?

Yes, there are other methods that can be used to prove that a Bessel function satisfies a given equation. For example, one can use the Frobenius method or the method of integral transforms to derive the Bessel function from the Bessel equation. Additionally, numerical methods can also be used to verify the accuracy of the Bessel function in satisfying a specific equation.

Can Bessel functions be used to solve real-world problems?

Yes, Bessel functions have numerous applications in real-world problems. For instance, they are used to describe the propagation of sound and heat in cylindrical or spherical systems, the behavior of electrical circuits, and the oscillations of a string or membrane. They are also used in signal processing, image processing, and other areas of science and engineering.

Similar threads

Finding 2 solutions of this Bessel's function using a power series
Replies
8
Views
2K
Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?
Replies
6
Views
1K
Question about the Frobenius method and Bessel functions
Replies
4
Views
2K
How should I use the Jacobi equation to determine the nature of this?
Replies
5
Views
1K
How to show that these sequences are summable?
Replies
1
Views
1K
How should I use the Jacobi equation to show this?
Replies
4
Views
1K
What is the Generating Function for Bessel Functions?
Replies
1
Views
2K
How should I find the equilibrium points and the general equation?
Replies
3
Views
589
How to get general solution via Green's function?
Replies
1
Views
1K
Prove that this series converges uniformly on R
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top