Side remark: Actually, to solve some linear ODE with polynomial coefficients like this, a general approach is to use the so-called Frobenius method: You start by assuming the solution can be written asxr?i=0?anxnThen you plug it back into the equation to determine r and to obtain some relations betweenan,an+1,an+2

The Legendre equations are the second order linear ODEs (1 -x2)y" - 2xy' + 1(1+ 1)y = 0 where I is some integer. 1. Verify that the Legendre polynomials, i.e. the orthonormal basis of the vector space of polynomials over [-1,1] that you get by doing Gram- Schmidt on the basis {1, x, x,x, using the inner product