How can I calculate j''(0)?

Question

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

Find j''(0) by using implicit differentiation

I got this for 0, but it was false. I want to know the process and answer of this as soon as possible.

Answer 1

J''(0)=-0.5

Answer 2

By taking x = 0 w get y'(0) =0.

Now we differentiate our equation: xy''' + y'' + y'' + y + xy' = 0; at x = 0 we get : 2y'(0)' + 1 = 0; y''(0) = - 1/2

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