Question

question below.

Answer 1

\[Let\ \ y= 1/(1+x^2).\]\[For\ any\ positive\ integer\ n,\ define\ f_{n}(x)=(1+x^2)^{n+1}y^{(n)}.\]\[Prove\ that\ f_{n+2}(x)+2(n+2)xf_{n+1}(x)+(n+1)(n+2)(1+x^2)f_{n}(x)=0\]