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I have been looking at series solutions to differential equations, and there is something that is not quite clear to me: namely, are ordinary points of equations necessarily ordinary points of solutions, and are singular points of solutions necessarily singular points of equations (and do the singularity types correspond)?

I have come across an example (perhaps it was Legendre's equation?) In which there was a regular singular point in the equation at x=0, yet the solution was fine here. I believe Fuch's theorem also stares that, at a regular singular point of an equation, at least one linearly independent solution exists (of a stated form) although I don't think it says anything about whether the solution is singular or not.

Perhaps I am not getting this because I don't quite understand what a singularity in an equation is. The coefficient of the highest derivative vanishes, and of the lower derivatives may or may not. I'm not entirely sure what this means for the solutions.

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