32.3. REGULAR SINGULAR POINTS 617

Example 32.3.2 Find the regular singular points of the equation and find the singularpoints.

x3 (x−2)2 (x−1)2 y′′+(x−2)sin(x)y′+(1+ x)y = 0

The singular points are 0,2,1. Let’s consider 0 first.

limx→0

x(x−2)sin(x)

x3 (x−2)2 (x−1)2

does not exist. Therefore, 0 is not a regular singular point. I don’t have to check any further.Now consider the singular point 2.

limx→2

(x−2)(x−2)sin(x)

x3 (x−2)2 (x−1)2 =18

sin2

andlimx→2

(x−2)2 1+ x

x3 (x−2)2 (x−1)2 =38

and so yes, 2 is a regular singular point. Now consider 1.

limx→1

(x−1)(x−2)sin(x)

x3 (x−2)2 (x−1)2

does not exist so 1 is not a regular singular point. Thus the above equation has only oneregular singular point and this is where x = 2.

Example 32.3.3 Find the regular singular points of

xsin(x)y′′+3tan(x)y′+2y = 0

The singular points are 0,nπ where n is an integer. Let’s consider a point at nπ wheren ̸= 0. To be specific, let’s let n = 3

limx→3π

(x−3π)3tan(x)xsin(x)

= 0

Similarly the limit exists for other values of n. Now consider

limx→3π

(x−3π)2 2xsin(x)

= 0

Similarly the limit exists for other values of n. What about 0?

limx→0

x3tan(x)xsin(x)

= 3

andlimx→0

x2 2xsin(x)

= 2

so it appears all these singular points are regular singular points.