Derivatives and Their Applications

State Leibnitz's theorem on higher order derivative. If

$$y^{\frac{1}{m}}+y^{-\frac{1}{m}}=2x$$

, then show that

a) $(x^2-1)y_2+xy_1-m^2y=0$

b) $(x^2-1)y_{n+2}+(2n+1)xy_{n+1}+(n^2-m^2)y_n=0$.

If $y=\log\left(x+\sqrt{a^2+x^2}\right)$, then show that $(a^2+x^2)y_{n+2}+(2n+1)xy_{n+1}+n^2y_n=0$

If $y=a\cos(\log x)+b\sin(\log x)$ then show that

i) $x^2y_2+xy_1+y=0$

ii) $x^2y_{n+2}+(2n+1)xy_{n+1}+(n^2+1)y_n=0$.