PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Special functions and orthogonal polynomials
Conditions of Validity, Ch Series solutions, singular points and where the special functions come from
Problem
Laguerre's equation with n=2 reads xy′′+(1−x)y′+2y=0, with a regular singular point at x=0.
((a)) Find the indicial equation and identify the exponent-difference case.
((b)) Construct the Frobenius series for that exponent, normalised so its first term is 1, and show that it terminates after the x2 term.
((c)) The second solution has the form y2=ylnx+∑k≥1bkxk, with y the polynomial of part (b). Find b1 and b2.