1. The problem statement, all variables and given/known data The bane of all physicists... 'Proof' questions... So we have the mapping, Δ : P3→P3 Δ[P(x)] = (x2-1) d2P/dx2 + x dP/dx And I need to prove that this is a linear mapping 2. Relevant equations Linear maps must satisfy: Δ[P(x+y)] = Δ[P(x)] + Δ[P(y)] and Δ[P(αx)] - αΔ[P(x)] 3. The attempt at a solution I'm not sure what to do. I've tried working through the actual mapping, performing the differential operations on the polynomial: ax3+bx2+cx+d and on: a(x+y)3+b(x+y)2+c(x+y)+d then expanded the brackets to see if I could separate the x and y terms to show that it's possible to pull them apart and demonstrate equality with Δ[P(x)] + Δ[P(y)] But it doesn't work. My best bet is that I've done something wrong regarding the part where I need to do d2P/d(x+y)2. I've never really had to work in that way before. Maybe I've gone wrong completely. Any advice? Thanks!