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# 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 $J'(0).$

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

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##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

in this problem where you've been the best of function of order zero. We know that this vessel punch a Why? Or Jeff Banks satisfies this given equation and mean over when it's zero vessel orders. You're easy. Book one. And we asked him point in part pages. So we're gonna blood zero in. We have for X. We have zero times J double finals zero plus J primary zero off. Zero Jeff. Zero physique zero. This is your This is your Oh, so we see that the engine premature is equal to zero in part, the rest on gentle, primal zero. So we're gonna, uh, take their with off this. Give any question that is ex. Let's do that. We have Why? Double prime plus extent, people prime. Plus why? Double crime? Why? Plus x times. Why Prime Music zero. We use Porter cool for the first time. And for the third term that we're gonna put zero in again. We have deejay double time of zero plus zero times. Jake Cipel Primary zero plus Jay. Double crime of zero plus Jeff. Zero plus zero Time J. Prime of zero is zero. We see that, Mr Zero yourself. Zero be noted. Jail zero or so. We have to J devil problem. Zero is equal to negative form. Promise? We see that, then. Jen. Double promise Jersey. Negative.

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