y=tan^-1x.
:. dy/dx=y_1=1/(1+x^2).
:. (1+x^2)y_1=1.
Re-differentiating w.r.t. x, using the Product Rule,
(1+x^2)(y_1)'+y_1(1+x^2)'=0.
:. (1+x^2)y_2+2xy_1=0.
:.((1+x^2)y_2)_n+(2xy_1)_n=0.............................(star).
Now, we use the following Leibnitz Rule (L) for the n^(th)
derivative of product of functions :
(uv)_n=u_n+""_nC_1u_(n-1)v_1+""_nC_2u_(n-2)v_2+...+v_n.
Applying L to (1+x^2)y_2," with, "u=y_2, and v=1+x^2,
((1+x^2)y_2)_n,
=(y_2)_n+n*(y_2)_(n-1)(2x)+(n(n-1))/2*(y_2)_(n-2)(2)+0,
=y_(n+2)+2nxy_(n+1)+n(n-1)y_n......(star^1).
Similarly, (2xy_1)_n,
=y_(n+1)+n*y_n*2+0..............................(star^2).
(star), (star^1), (star^2)rArr y_(n+2)+(2nx+1)y_(n+1)+n(n+1)y_n=0.
