Question

show that y sub n=cos(ncos^-1(x)) satisfies the differential equation:
(1-x^2)(d^2y/dx^2)-x(dy/dx)+n^2(dy/dx)=0

Answer 1

seems pretty straightforward:

y = cos(n arccos(x))
y' = n sin(n arccos(x)) / √(1-x^2)
y'' = nx sin(n arccos(x))/(1-x^2)^(3/2) - n^2 cos(n arccos(x))/(1-x^2)

(1-x^2) y'' = nx sin(n arccos(x))/√(1-x^2) - n^2 cos(n arccos(x))

(1-x^2) y'' - xy' = n^2 cos(n arccos(x))

(1-x^2) y'' - xy' - n^2 y' doesn't make it go to zero.

That last step doesn't seem right. Is there a typo? Just the way it's shown doesn't seem right; two separate terms with y'