problem 11.1

Bessel s differential equation

Bessel s differential equation of the nth order is given by

where n = 0 , 1 , 2 , (zero or positive integers). Show that this has a following power series solution A( r) = J _n( r):

where 0! := 1, 1! := 1, and k! := 1 2 k for an integer k > 0. [Note: Equation (11.29) corresponds to n = 1.]

problem 11.2

Axisymmetric Gross Pitaevskii equation

Axisymmetric Gross Pitaevskii equation is given by (11.31):

Show that, (i) for small ?, ( : a constant); (ii) for large ? ( ? 1), = 1 ? 1/2 ? ^?2 + .

problem 11.1

: We write the power series solution beginning from r ^p ( p: a positive integer) as . Substituting this in (11.32), the lowest-power term is ( p ² ? n ²) r ^{p ?2}. From (11.32), we must have p = n. Thus, we set p = n. Then, we have , . Substituting these into (11.32) and collecting the terms including r ^{k ?2}, we obtain

Hence we find the relation between neighboring coefficients as

If we set the first coefficient as c _n = 1 /(2 ⁿ n!), we have

These are consistent with the first...