Consider the equation x2y″ + xy′ + (1 − x)y = 0. (a) Show that its exponents are ±i, so it...

Consider the equation x2y″ + xy′ + (1 − x)y = 0. (a) Show that its exponents are ±i, so it has complex-valued Frobenius series solutions

with p0 = q0 = 1. (b) Show that the recursion formula is

Apply this formula with r = i to obtain pn = cn, then with r = −i to obtain qn = cn. Conclude that pn and qn are complex conjugates: pn = an + ibn and qn − an − ibn where the numbers [an] and [bn] are real. (c) Deducefrom part (b) that the differential equation given in thisproblem has real-valued solutions of the form

where