Answer to Question #126166 in Differential Equations for Ayesha

Question #126166
Regular singular point for differential equationx(x-2)²d²y/dx²+2(x-2)dy/dx+(x+3)y=0
1
Expert's answer
2020-07-13T18:54:25-0400

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.

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