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Problem 79 Hard Difficulty

The Bessel function of order 0, $ y = J (x), $ satisfies the differential equation $ xy" + y' + xy = 0 $ for all values of $ x $ and its value at 0 is $ J(0) = 1. $
(a) Find $ J'(0). $
(b) Use implicit differentiation to find $ J'(0). $

Answer

(a) $$
J^{\prime}(0)=0
$$
(b) $$
J^{\prime \prime}(0)=-\frac{1}{2}
$$

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Video Transcript

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