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

[ductape]

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 seperately? I suspect that I do I just want to clarify.
Thanks

 

[matematikawan]

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.

 

[HallsofIvy]

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.

 

[matematikawan]

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.
Is this what ductape meant in the initial post? Used two separate series.