Question

Find all singular points of the following equation and determine whether each one is regular or irregular?

x? (1 – x)2y" + (x - 1) y' + 4y = 0

a) 0 and 1 are regular singular points.

b) 1 is a regular singular point, 0 is an irregular singular point.

c) 0 is a regular singular point, 1 is an irregular singular point.

d) none of these

e) 0 and 1 are irregular singular points.

Answer 1

SOIR For a differential equation of the follown following form Ten) et P, (1) f'(a) + Po (n) f(m1=0 e is said to be (A point as is an or ordinary point if functions Pi (2) and Po (n) one analytic at naa, otherwise singular point. A point- c is a regular singular point if liens has a pole up to order 1 at nac and Po(a) has a pole of order up to 2 at nas () A singular point a which is not a a regular singular point is called an inegular singular peint, How given equation is & (Amary" + (mrily + 4y=0 This can be written as 1 yht Y't 4 (inn) Ý 20 (1-3) 22 2 Compaining with wegel- P, (01) = Poln) 22 (1-1) 1 n (1-9

Thus, I is a regular singular point and o is an in inregular singular point, 6 So aption (6) is connect in the given question, OLD GOL 2 way ON
Correction:

Here by mistake I forgot to put one minus sign before the expression of p1(x). That is where I have used p1(x), it should be -p1(x) not p1(x). Since existance of limit and analyticity does not depend on the multiplication by a constant, so the solution is correct once you assume a minus sign before p1(x) in the solution.

So keep it in mind that there is a minus sign before the expression of p1(x).

Thank you!!