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Find all singular points?

Find all singular points of xy"+(1-x)y'+xy=0 and determine whether each one is regular or irregular.

Answer: x=0, regular

I know that x=0 but I need to take the limit as x approaches to 0 but what function should I take the limit?

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Hassan H. | Math Tutor (All Levels)Math Tutor (All Levels)
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Hello Sun,

Recall that a singular point x = x0 of an equation such as

P(x)y'' + Q(x)y' + R(x)y = 0

is called a regular singular point if both of the following are analytic at x0:

(x - x0)( Q(x)/P(x) ) and (x - x0)2( R(x)/P(x) ).

Since your equation has P(x) = x, Q(x) = 1-x, and R(x) = x, and these are all polynomials, checking that the above quotients are analytic at x0 amounts to seeing whether their limits are finite as you x → x0.  That is, compute the limits

limx → 0 (x(1-x)/x) and limx → 0 (x(x)/x)

and see if they are finite (they are).  If so, the point x = 0 is a regular singular point.

Hope this clears things up.


Hassan H.