   Chapter 2.6, Problem 61E

Chapter
Section
Textbook Problem

# The Bessel function of order 0, y = J ( x ) satisfies the differential equation x y ′ ′   +   y ′ +   x y   =   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).

To determine

(a)

To Find:

J(0)

Explanation

1) Concept:

Use implicit Differentiation. Differentiation with respect to x, y must be treated as a function of x.

2) Calculations:

We have,

The Bessel function of order 0,y=Jx

Since Jx satisfies the given differential equation,

xy''+y'+xy=0

y=J(y)

y'=J'(y)

y''=J''y

Substitute these values in given differential equation,

xJ''(x)+J'(x)+x <

To determine

(b)

To find:

J''0

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