If 2x = y ^ (1)/(5) + y ^ - (1)/(5) and ( x ^ 2 - 1) d ^ 2 y d x ^...

Given ${y}^{\dfrac{1}{5}} + {y}^{- \dfrac{1}{5}} = 2x$

$⇒ \left(\dfrac{1}{5}{y}^{- \dfrac{4}{5}} - \dfrac{1}{5}{y}^{- \dfrac{6}{5}}\right)\dfrac{dy}{dx} = 2$

$⇒ {y}^{′}\left({y}^{\dfrac{1}{5}} - {y}^{- \dfrac{1}{5}}\right) = 10y$

$⇒ {y}^{′}\left(2\sqrt{{x}^{2} - 1}\right) = 10y$

$⇒ {y}^{′}\sqrt{{x}^{2} - 1} = 5y$

Differentiating again

${y}^{′′}\left(2\sqrt{{x}^{2} - 1}\right) + {y}^{′}2\dfrac{2x}{2\sqrt{{x}^{2} - 1}} =10y'$

$⇒ {y}^{′′}\left({x}^{2} - 1\right) + x{y}^{′} = 5\sqrt{{x}^{2} - 1}\left({y}^{′}\right)$

$⇒ {y}^{′′}\left({x}^{2} - 1\right) + x{y}^{′} - 25y = 0$

$⇒ \lambda = 1, k = - 25$