Answer to Question #126166 in Differential Equations for Ayesha

Question

Regular singular point for differential equationx(x-2)²d²y/dx²+2(x-2)dy/dx+(x+3)y=0

Answer 1

Solution

Let us simplify the discussion only to the second order differential equations in the standard form of y′′+P(x)y′+Q(x)y=0.

x(x-2)²y"+2(x-2)y'+(x+3)y=0

First let us bring DE into standard form (1).

y"+2/(x(x-2))y'+(x+3)/(x(x-2)²)y=0

Two singularities are observed at x={0,2}

1. According to the rule, for the singularity at x=0, we have to multiply P(x)=2/(x(x-2)) by x and Q(x)=(x+3)/(x(x-2)²) by x². Then both members are analytic so the DE is a regular singularity at x=0
2. For the case at x=2, if we multiply P(x) by (x−2) and Q(x) by (x−2)², these members become analytic. So at x=2 DE is regular singularity.