Determine the singular points of the given differential...

Determine the singular points of the given differential equation. Classify each singular point as regular or irregular.
$$x^{2}(x-5)^{2} y^{\prime \prime}+4 x y^{\prime}+\left(x^{2}-25\right) y=0$$

Key Concepts

Singular Points

These are points in the domain of a differential equation where the coefficient of the highest derivative either vanishes or fails to be analytic. Recognizing singular points is crucial because the nature of the solution can change dramatically at these points, requiring special techniques for analysis.

Ordinary Points

These are points where every coefficient in the differential equation is analytic and the leading coefficient (i.e., the coefficient of the highest derivative) is nonzero. At ordinary points, standard power series methods can reliably provide analytic solutions.

Regular Singular Points

A singular point is classified as regular if, upon re-writing the differential equation in a standard form, the coefficients of the lower order derivatives have poles of order at most one or two (depending on the order of the derivative) at that point. This classification is important because it allows the use of the Frobenius method to find solutions.

Irregular Singular Points

Irregular singular points are those where the coefficients, when normalized by the leading coefficient, exhibit more severe singularities (poles of higher order or essential singularities) than permitted for a regular singular point. Such points are more challenging to analyze and typically do not permit series solutions via the Frobenius method without modifications.

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