## gerryliyana 2 years ago Second Order Differential Equations

1. gerryliyana

$y'' + \frac{ 1 }{ x }y'+ \left( 1-\frac{ n^{2} }{ x^{2} } \right)y = 0$ how much the solution zero-order of the Bessel equations above at (n=0) in between intervals of π along the positive x axis"????

2. kryton1212

@ajprincess would you kindly help my brother if you are free....

3. ajprincess

I am extremely sorry @kryton1212.:( I havnt learnt differential equations yet.

4. kryton1212

5. kryton1212

if they learnt it *

6. ajprincess

@hartnn, @experimentX, @unklerhaukus, @apoorvk. Can u all plz help with this question?

7. kryton1212

hartnn is not online yet...

8. ajprincess

But @unklerhaukus and @apoorvk are online. They may be able to help.:)

9. kryton1212

thank you @ajprincess :)

10. ajprincess

Welcome:)

11. kryton1212

12. UnkleRhaukus

can you reword the question, im not sure what you are asking

13. kryton1212

@sauravshakya would you kindly help my brother if you are free?

14. gerryliyana

@UnkleRhaukus " how much the minimum solution of the equations above (at n=0) in between intervals of π along the positive x axis????"

15. UnkleRhaukus

i dont know what you mean by "how much the ..... solution?" are you looking for the number of solutions? are you looking for the solution? are you looking for the value of the function at one of the solutions?

16. gerryliyana

i'm looking for the least number of solutions..,

17. gerryliyana

the minimum number of solutions

18. UnkleRhaukus

ah ok.

19. experimentX

write it up as $x^2 y'' + xy' + (x^2-n^2)y = 0$ this is a standard non liner differential equation called Bessel differential equation http://en.wikipedia.org/wiki/Bessel_function http://mathworld.wolfram.com/BesselDifferentialEquation.html use power series method to find it's solution

20. robtobey

Even though I'm 78 years old and cannot integrate x with pencil and paper, I've access to the computer program, Mathematica 8 Home Editon. that can. The attachment shows the general solution with two constants of integration. A plot of the solution from x = 0 through x = 6 Pi is included.