MHB Find nth Derivative - Quick Tutorial

[anil86]

Find nth derivative:

Please view attachment!View attachment 1701

 

[Ackbach]

Could you please re-scan your attachment? The current one is so out-of-focus as to be nearly useless.

 

[anil86]

Could you please re-scan your attachment? The current one is so out-of-focus as to be nearly useless.

I regret for the inconvenience. View attachment 1716

 

[Opalg]

The assignment does not ask you to find the $n$th derivative, but to prove that $$(1-x^2)y_{n+1} - 2(\gamma + nx) y_n -n(n-1)y_{n-1} = 0.$$

You have shown that $$y_1 = \frac{\gamma y}{1+x} + \frac{\gamma y}{1-x} = \frac{2\gamma y}{1-x^2}.$$ Write that as $$(1-x^2)y_1 - 2\gamma y = 0.$$ Differentiate, to get $$(1-x^2)y_2 - 2xy_1 - 2\gamma y_1 = 0.$$ Now use that as the base case for a proof by induction.