Additional solutions of a weighted heat equation (Q654226)

Consider the differential equation \[ {d\over dx}\Biggl[(1- x^2){dv\over dx}\Biggr]= -\lambda^2 v.\tag{\(*\)} \] In case \(\lambda^2= n(n+1)\), \((*)\) represents the Legendre's differential equation. Using this fact, the author describes a method to calculate infinitely many solutions to \((*)\).