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nth derivative

anil86

New member
Nov 1, 2013
10
Find nth derivative:

Please view attachment!!!Image0354.jpg
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Could you please re-scan your attachment? The current one is so out-of-focus as to be nearly useless.
 

anil86

New member
Nov 1, 2013
10
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. Image0356.jpg
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
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.