nth derivative

anil86

New member
Find nth derivative:

Ackbach

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

anil86

New member
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.

Opalg

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

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