Finding 2 solutions of this Bessel's function using a power series

happyparticle
Messages
490
Reaction score
24
Homework Statement
find 2 solutions of this Bessel's function using a power series.
Relevant Equations
##x^2 d^2y/dx^2 + x dy/dx + (x^2 -9/4)y = 0##
I have to find 2 solutions of this Bessel's function using a power series.

##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0##

I'm using Frobenius method.

What I did so far

I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##

I got ##\sum[(n+p)(n+p-1)a_n x^{n+p}] + \sum[(n+p)a_n x^{n+p}] + \sum[a_n x^{n+p+2}] - 9/4 \sum[a_n x^{n+p}] = 0##

Thus, the smallest power for n=0 is ##x^p##, so I found ##p = \pm 3/2##

To have ##x^{m+p}##

I got ##(m+p)(m+p-1)a_m + (m+p)a_m + a_{m-2} - 9/4 a_m = 0=> a_{m-2} = [(m+p(m+p-1) + (m+p) -9/4)] a_m##

For p = 3/2 I have ##a_{m-2} = -m(m+3)a_m##

So far I'm stuck here, since I can't find a series with this above and then by plugging the solution in the standard form I should get some cos and sin from the hint. I guess I made an mistake somewhere but I don't know where.

Any help will be really appreciate.
 
Last edited by a moderator:
Physics news on Phys.org
Is that a typo in the equation? Do you mean the second term to be ##x (dy/dx)##?
 
FactChecker said:
Is that a typo in the equation? Do you mean the second term to be ##x dy/dx##?
yes, my bad.
 
EpselonZero said:
Homework Statement:: find 2 solutions of this Bessel's function using a power series.
Relevant Equations:: ##x^2 d^2y/dx^2 + x dy/dx + (x^2 -9/4)y = 0##

I have to find 2 solutions of this Bessel's function using a power series.

##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0##

I'm using Frobenius method.

What I did so far

I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##

I got ##\sum[(n+p)(n+p-1)a_n x^{n+p}] + \sum[(n+p)a_n x^{n+p}] + \sum[a_n x^{n+p+2}] - 9/4 \sum[a_n x^{n+p}] = 0##

Thus, the smallest power for n=0 is ##x^p##, so I found ##p = \pm 3/2##

What do you get if you set n = 1? Does that restrict a_1, or are you free to choose it as you wish?

For n \geq 2 I agree that you have <br /> a_{n}\left(n(n+2p)\right) + a_{n-2} = 0 where p = \pm\frac32.

The form of the recurrence allows you to treat the even terms n = 2k and the odd terms n = 2k+1 separately. This leads you to linear recurrences of the form <br /> b_{k} = f(k)b_{k-1} where b_{k} = a_{2k} or a_{2k+1} as appropriate. In solving these it may assist you to note that <br /> \prod_{r=0}^{k-1}(r + p) = \frac{\Gamma(k+p)}{\Gamma(p)} where \Gamma is the Gamma function.

So far I'm stuck here, since I can't find a series with this above and then by plugging the solution in the standard form I should get some cos and sin from the hint. I guess I made an mistake somewhere but I don't know where.

Any help will be really appreciate.

Please post the hint.
 
pasmith said:
Please post the hint.
Using the standard form and my solutions I should get
##\sqrt{\frac{2}{\pi x}} (\frac{sin x}{x} - cos x)## and ##-\sqrt{\frac{2}{\pi x}} (\frac{cos x}{x} + sin x)##

I'm not sure to understand how you get ##
b_{k} = f(k)b_{k-1}
##

I though by adding some terms to ##a_{m-2} = -m(m+3)a_m## I could find the serie.
 
EpselonZero said:
Using the standard form and my solutions I should get
##\sqrt{\frac{2}{\pi x}} (\frac{sin x}{x} - cos x)## and ##-\sqrt{\frac{2}{\pi x}} (\frac{cos x}{x} + sin x)##

The general solution of <br /> x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + \left(x^2 - \frac94\right)y = 0<br /> is y(x) = AJ_{3/2}(x) + BJ_{-3/2}(x). Now it happens that \begin{split}<br /> j_1(x) &amp;= \frac{\sin x}{x^2} - \frac{\cos x}{x} = \sqrt{\frac{\pi}{2x}} J_{3/2}(x) \\<br /> y_1(x) &amp;= -\frac{\cos x}{x^2} - \frac{\sin x}{x} = \sqrt{\frac{\pi}{2x}} J_{-3/2}(x)<br /> \end{split} where j_1 and y_1 are spherical bessel functions, which are the solutions of <br /> x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - 2)y = 0. So the solutions given in your hint are <br /> \begin{split}<br /> \sqrt{\frac{\pi x}{2}}j_1(x) &amp;= \frac{\pi}{2} J_{3/2}(x),\mbox{ and} \\<br /> \sqrt{\frac{\pi x}{2}}y_1(x) &amp;= \frac{\pi}{2} J_{-3/2}(x).\end{split}

You have yet to find the series for J_{\pm3/2}. Once you have found them, you can attempt to recognise them as the series of the functions given in the hint. To that end, it is best to use <br /> \prod_{k=0}^{n-1} \left(k + \frac32 \right) = \frac{1}{2^n} \prod_{k=0}^{n-1} (2k + 3) rather than the expresssion involving Gamma functions.

Go back and look again at the result of applying Frobeniu's method. Setting n = 0 told you the possible values of p, because a_0 is constrained to be non-zero. What happens if you set n = 1? Does that tell you anything about the possible values of a_1?

Have you not encountered first order linear recurrences before? It surprises me - although perhaps it no longer should - that they are not regularly taught alongside series solution of differential equations, since you can't really do one without the other.
 
I'm following the method used in the textbook mathematics for physicist by lea.
Here's the example I'm following. Maybe you'll understand what I'm trying to do.
If I'm correct I found the recursion relation and then I plugged p = 3/2.

I found ##a_m = -\frac{1}{(m)(m+3)} \cdot -\frac{a_{m-4}}{(m-2)(m+2)}##
##a_m = \frac{-1^{m/2 + 1}}{(m+2)!(m+5)!}a_0##
Should I find the odd solution and add to the even solution to have the full solution for p = 3/2 ?
4iFJUyn.png
 
Last edited by a moderator:
From ##a_m = \frac{-1^{m/2}}{m!(m+3)!} a_0##
I found for m even
##a_{2k} = \frac{-1^{k}}{2k!(2k+3)!} a_0##

Now I'm wondering if I can find the odd series using only the recurrence relation and can I add up both series?
Furthermore, I don't see any cos or sin series yet.
 
Last edited by a moderator:
I think I'm not that far since I guess a solution should be in this form ##x^{1/2} \sum\limits_{n = 0}^\infty {a_n x^{2n} } ## but I don't see how to have (2n+1)! to the dominator.
 
Back
Top