Can Laplace Transforms Reveal Bessel Functions as Solutions?

In summary: I'm just not sure what to do with my s^2+1. Can I write that in terms of a Bessel function?In summary, the conversation discusses the use of Laplace transforms to solve an equation involving a second derivative and initial conditions. The solution is shown to be the Bessel function of order zero, but the poster is unsure of how to prove this without a reference table. They mention trying to take the Laplace transform of xy'' and getting a "dumb answer," but realize that they need to take into account that Y(s) is a function of s when doing derivatives. They ask for help on proving the solution without a reference table.
  • #1
Crush1986
207
10

Homework Statement


[tex] xy''+y'+xy=0, y'(0)=0, y(0)=0 [/tex] Using the method of Laplace transforms, show that the solution is the Bessel function of order zero.

Homework Equations


[tex] -(d/ds)L{f(x)} [/tex]

The Attempt at a Solution


The only thing I got out of this when trying to solve it was y=0. Obviously not the intended answer. In the problem I'm told that the answer is the Bessel function of 0 order.

I'm pretty sure the parts I'm messing up is the Laplace transform of xy''. I haven't tried to take the Laplace transform of anything like that before and I'm sure it is where I'm messing up. I tried to use the relevant equation up there and I got...

[tex] -(d/ds)[(s^2Y(s)-s)]+sY(s)-1+-(d/ds)[Y(s)]=0[/tex]

This gives me [tex] -2sY(s)+1+sY(s)-1=0 [/tex]

=> [tex] -sY(s)=0 [/tex]

which yeah, just gives me a dumb answer. Again I'm pretty sure it's with my xy'' term. Maybe even more :(

Thanks anyone for help.
 
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  • #2
Oh... as soon as I posted this I thought... Y(s) is a function of s... I need to take that into account when doing derivatives... Dang it... as always it's right after I post I have a decent idea...
 
Last edited:
  • #3
Ok so I'm on the right track for sure. My current expression is

[tex] y = L^-1[1/(s^2+1)^.5] [/tex]

I know that the inverse of that is the zero order Bessel function. How would I prove with no table though?
 
  • #4
Crush1986 said:
Ok so I'm on the right track for sure. My current expression is

[tex] y = L^-1[1/(s^2+1)^.5] [/tex]

I know that the inverse of that is the zero order Bessel function. How would I prove with no table though?
Well fortunately, there are tables of the Laplace transforms of all kinds of special functions, including Bessel functions. One reference which has such a table is Abramowitz & Stegun, Handbook of Mathematical Functions, p. 1029:

http://people.math.sfu.ca/~cbm/aands/

If you don't trust such a reference, I suppose you could always derive the Laplace transform of your Bessel function from the definition and compare it with your earlier result.
 
  • #5
SteamKing said:
Well fortunately, there are tables of the Laplace transforms of all kinds of special functions, including Bessel functions. One reference which has such a table is Abramowitz & Stegun, Handbook of Mathematical Functions, p. 1029:

http://people.math.sfu.ca/~cbm/aands/

If you don't trust such a reference, I suppose you could always derive the Laplace transform of your Bessel function from the definition and compare it with your earlier result.
Yah I'm supposed to derive it I'm sure.
 

1. What is a Laplace Transform?

A Laplace Transform is a mathematical tool used to convert a function from the time domain to the frequency domain. It is denoted by the symbol "L" and is often used in engineering and physics to solve differential equations.

2. How do I find the Laplace Transform of a function?

To find the Laplace Transform of a function, you can use the formula L[f(t)] = ∫0∞ f(t)e-st dt, where "s" is a complex variable. You can also use tables or software programs to calculate the transform.

3. What is the purpose of a Laplace Transform?

The Laplace Transform is used to simplify and solve differential equations by converting them into algebraic equations in the frequency domain. This makes it easier to analyze and understand the behavior of systems and signals.

4. Can a Laplace Transform be inverted?

Yes, a Laplace Transform can be inverted using the inverse Laplace Transform. This converts a function from the frequency domain back to the time domain. The inverse transform is denoted by the symbol "L-1" and is calculated using a similar formula as the Laplace Transform.

5. What are the common applications of Laplace Transforms?

Laplace Transforms are commonly used in electrical engineering, control systems, and signal processing. They can also be applied in various fields such as physics, economics, and biology to model and analyze systems and phenomena.

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