- #1

happyparticle

- 443

- 20

- 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.

##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.

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