math (differential equation)

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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

  • math (differential equation) -

    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'

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