Consider the differential equationy+ xs y+ xty = 0,

Chapter 5, Problem 21

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Consider the differential equationy+ xs y+ xty = 0, (i)where = 0 and = 0 are real numbers, and s and t are positive integers that for themoment are arbitrary.(a) Show that if s > 1 or t > 2, then the point x = 0 is an irregular singular point.(b) Try to find a solution of Eq. (i) of the formy = n=0anxr+n, x > 0. (ii)Show that if s = 2 and t = 2, then there is only one possible value of r for which there is aformal solution of Eq. (i) of the form (ii).(c) Show that if s = 1 and t = 3, then there are no solutions of Eq. (i) of the form (ii).(d) Show that the maximum values ofs and t for which the indicial equation is quadratic inr [and hence we can hope to find two solutions of the form (ii)] are s = 1 and t = 2. Theseare precisely the conditions that distinguish a weak singularity, or a regular singularpoint, from an irregular singular point, as we defined them in Section 5.4.As a note of caution, we point out that although it is sometimes possible to obtain a formalseries solution of the form (ii) at an irregular singular point, the series may not have apositive radius of convergence. See for an example.