The result x 2 u " + [ x 2 + 1 4 − v 2 ] u = 0 by using the given data. Formula used: The Bessel’s equation is x 2 y " + x y ' + ( x 2 − v 2 ) y = 0 and its solution is y " + p ( x ) y ' + q ( x ) y = 0 . Calculation: The Bessel’s equation is given as x 2 y " + x y ' + ( x 2 − v 2 ) y = 0 . Divide x 2 throughout the equation x 2 y " + x y ' + ( x 2 − v 2 ) y = 0 . y " + 1 x y ' + ( 1 − v 2 x 2 ) y = 0 Compare the equation y " + 1 x y ' + ( 1 − v 2 x 2 ) y = 0 with y " + p ( x ) y ' + q ( x ) y = 0 here p ( x ) = 1 x and q ( x ) = 1 − v 2 x 2 . Differentiate the equation p ( x ) = 1 x with respect to x . d d x ( p ( x ) ) = d d x ( 1 x ) P ' ( x ) = − 1 x 2 Elimination of first derivative is given as v ( x ) = exp ( − 1 2 ∫ p ( x ) d x ) . Substitute the value of p ( x ) = 1 x in v ( x ) = exp ( − 1 2 ∫ p ( x ) d x ) . v ( x ) = exp ( − 1 2 ∫ 1 x d x ) v ( x ) = exp ( − 1 2 ln x ) = exp ( ln x − 1 2 ) = x − 1 2 From the problem 16 the resultant equation is u " + u ( q ( x ) − 1 4 P ( x ) 2 − 1 2 P ' ( x ) ) = 0 . Substitute the value of P ( x ) , P ' ( x ) and Q ( x ) in u " + u ( q ( x ) − 1 4 P ( x ) 2 − 1 2 P ' ( x ) ) = 0 . u " + u ( − 1 4 ( 1 x ) 2 − 1 2 ( − 1 x 2 ) + 1 − v 2 x 2 ) = 0 u " + u ( − 1 4 x 2 + 1 2 x 2 + 1 − v 2 x 2 ) = 0 u " + u ( 1 − v 2 x 2 + 1 4 x 2 ) = 0 x 2 u " + u ( x 2 − v 2 + 1 4 ) = 0 Hence, the result x 2 u " + [ x 2 + 1 4 − v 2 ] u = 0 is proved.

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