I know that legendre polynomials are solutions of the differential equation is (1-x^2)d^2y/dx^2 - 2x dy/dx+l(l+1)y=0 where l is an integer. The first five solutions are P0(x)=1, P1(x)=x, P2(x)=3/2x^2-1/2, P3(x)=5/2x^3-3/2x, P4(x)=35/8x^4-15/4x^2+3/8 The problem is that I don't understand what the problem is telling me to do. It says to show that each of the polynomials Pl(x) solves the differentil equation with its particular value l. Do I just plug in l? For example, for P0(x)=1, would I plug in 1 for x and 0 for l? I'm really confused. Another problem is that I have to show by doing 10 integrals that if l is not equal to m, that integral from -1 to 1 dxPl(x)Pm(x)=0 so that these polynomials are orthogonal on the interva1 [-1,1]. Do I just take a value for l and one for m 10 times. So for the first integral, m=1 and n=2?