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Homework Statement
Consider
[itex] x^2y''-xy'+n^2y=0 [/itex]
where n is a constant.
a) find two linearly independent solutions in the form of a Frobenius series, initially keeping at least the first 3 terms. Can you find the solution to all orders?
b) for n=1 you shouild find only one linearly independent solution. If y_1(x) and y_2(x) are two independent solution for n=/= 1, us the fact that
[itex]\tilde{y} = [1/(n-1)][y_1(x) - y_2(x)][/itex]
is a solution and take the [itex] n \rightarrow 1[/itex] of it to find another solution.
Homework Equations
The Attempt at a Solution
I did the normal method of frobenius by taking the maclaurin expansion of y(x) and plugging it back in and got
[itex]y(z) = \sum_{k = 0}^{ \infty} a_kx^{(k+s)}[n^2-(k+s)+(k+s)(k+s-1)] = 0[/itex]
and by asserting that a_0 does not equal 0 i solved for s and got
[itex]s = 1 \pm \sqrt{1-n^2}[/itex]
But all the examples i have looked at have had another summation that starts at a different value of k in order to get a recurrence relationship between the coefficients. I only have one summation so i don't know what to do.
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