Converting 2nd order ODE to Bessel Function

In summary, the conversation discusses solving a second order differential equation using the generalized solution to Bessel's equation. The original equation is manipulated by multiplying both sides by x and dividing by x in order to solve the resulting Bessel function. The question is then raised about dividing by x to simplify the equation, but it is determined that this would not work due to x still multiplying the second derivative. The conversation ends with a request for help in converting the equation further.
  • #1
rjg6
2
0

Homework Statement


I am attempting to solve the 2nd order ODE as follows using the generalized solution to the Bessel's equation


Homework Equations


original ODE:
x[tex]d^{2}y/dx^{2}[/tex]-3[tex]dy/dx[/tex]+xy=0

The Attempt at a Solution


My first thought is to bring out an x^-1 outside of the function so that I end up with:
[tex]x^{-1}([/tex][tex]x^{2}[/tex][tex]d^{2}y/dx^{2}[/tex]-3x[tex]dy/dx[/tex]+[tex]x^{2}[/tex]y)=0
I would then solve the resulting Bessel equation found inside the parentheses, and multiply the resulting solution by x^-1. Is this at all a legal operation? Thank you.
 
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  • #2
There is no need for the x-1 outside. Just multiply both sides of the original equation by x.
 
  • #3
Now if the opposite were true and I was trying to drop the power of x's by 1:

example: (x^3)[tex]d^{2}y[/tex]/[tex]dx^{2}[/tex]+(x^2)dy/dx+(x^3)y=0

Could I then instead divide by x to come up with:
(x^2)[tex]d^{2}y[/tex]/[tex]dx^{2}[/tex]+(x)dy/dx+(x^2)y=0

with the understanding that the solution to the resulting Bessel function would exclude any results for when x-> 0?
 
  • #4
Well, because x2 still multiplying the second derivative that would be a problem any way, but you are right if you divided by something that completely got rid of a function multiplying the highest derivative, then you would have to add that condition.
 
  • #5
Hey I need some help in converting the second order differential equation into..
I was able to convert the original equation into the following form x^2*y''+2x*y'+x^2*y=0
I am not able to move forward from here..
Please could you suggest some method
 

What is a 2nd order ODE?

A 2nd order ordinary differential equation (ODE) is an equation that contains a second derivative of a function with respect to an independent variable. It can be written in the form: d2y/dx2 = f(x,y,dy/dx), where x is the independent variable, y is the dependent variable, and f is a function of x, y, and the first derivative dy/dx.

What is a Bessel function?

A Bessel function is a special type of mathematical function that arises in many areas of physics and engineering. It is defined as the solution to the Bessel differential equation, which has the form: x2d2y/dx2 + xdy/dx + (x2 - α2)y = 0, where α is a constant. Bessel functions are important in the study of wave phenomena, such as heat transfer and sound propagation.

Why do we need to convert a 2nd order ODE to Bessel function?

Converting a 2nd order ODE to Bessel function can be useful in solving certain types of differential equations. Bessel functions have many properties that make them easier to work with, such as orthogonality and recurrence relations. In some cases, converting to Bessel functions can simplify the problem and lead to a more elegant solution.

What is the process for converting a 2nd order ODE to Bessel function?

The process for converting a 2nd order ODE to Bessel function involves substituting the Bessel solution y(x) = u(x)v(x) into the ODE and solving for u(x) and v(x). This results in two separate equations, one for u(x) and one for v(x), which can then be solved using known properties of Bessel functions. The solutions are then combined to find the general solution to the original ODE.

What are the applications of converting a 2nd order ODE to Bessel function?

Converting a 2nd order ODE to Bessel function has many applications in physics and engineering. Some examples include solving problems in heat transfer, fluid mechanics, electromagnetism, and quantum mechanics. Bessel functions are also used in the analysis of vibrating systems, such as drums and guitar strings. In general, they are important tools in solving problems that involve wave-like behavior.

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