Question

7.6.26

7.6.26 (a) Show that has two solutions: (b) For α 0 the two linearly independent solutions of part (a) reduce to the single solution y o. Using Eq. (7.68) derive a second solution, Verify that y is indeed a solution. (c) Show that the second solution from part (b) may be obtained as a ing case from the two solutions of part (a): lim ( yi-32 Y2 (x) α-0

It's in Mathematical Methods for Physicists 7e, Arfken

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Answer 1

1-α (a) y" +-y = 0 (1) 4x We check for y1 (x) = aoX(1+α)/2 1+α Then yi,(x) + 4x2 -y,(x) 1+α aox 2 2 -3+α 4 4 Hence (1) is satisfied aoX(1-a)/2 Next we check for ½(x) y, (x) =

Then aox 2 4x 4 4 Hence (1) is satisfied Thus y1(x) ,½(x) are solutions of (1) 0, then the two linearly independent solutions reduce b) When α to one solution y1,(x) = aoX1/2. For the other solution we let

Then 4 Substituting these values in (1), while noting that a 0, we get 4x2 1 a 4

Thus we got u',x5 + u,x--= 0 Integrating both sides, we get or, in u' =-In x or,u Further integration, gives u=In x Hence the second linearly independent solution is c) 1+α )-)lim s lim

Applying L'Hospital rule we find ao51+α(2) In x-aoxT-1)In x = aoX1/2 (ln x) (proved) 1-α = lim α-0