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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.