Is the Frobenius Method Required at Each Singular Point for General Solutions?

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ductape
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Hello, I was just wondering, if I have a differential equation that has two regular singular points, and I am asked for the general solution, do I need to use the Frobenius method about each point separately? I suspect that I do I just want to clarify.
Thanks
 
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I think you have the option even not to use the Frobenius method. Can we avoid the regular point?

The Legendre's equation
(1-x2)y" - 2xy' + n(n+1)y = 0
have two regular points but we usually solve the equation about the ordinary point x=0.
 
Typically, we have to use Frobenious method at singular points only when we are given intial conditions at that singular point. It is easier to apply "y(x0)" and "y'(x0)" if our series solution is in terms of (x- x0)n.
 
Yes it looks reasonable to used frobenius series for an initial value problem because of an infinite number of terms involve.

But what about a boundary value problem where we are given two points? y(x0) = y0 and y(x1) = y1 .

We still need to sum an infinite series. :confused:
Is this what ductape meant in the initial post? Used two separate series.