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(i) If y^(1//m) + y^(-1//m) = 2x then pr...

(i) If `y^(1//m) + y^(-1//m) = 2x` then prove that ` (x^(2)-1) (d^(2)y)/(dx^(2)) + x (dy)/(dx) - m^(2)y =0`
(ii) If y = ln `(x + sqrt(1+x^(2))` , then prove that ` (1+x^(2)) (d^(2)y)/(dx^(2)) + x (dy)/(dx)=0`

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To solve the given problem, we will break it down into two parts as specified in the question. ### Part (i) Given the equation: \[ y^{\frac{1}{m}} + y^{-\frac{1}{m}} = 2x ...
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