(a) Show that the Bessel function J0(x) given by Formula(4) of Section 9.8 satisfies the

Chapter 9, Problem 45

(choose chapter or problem)

Show that the Bessel function \(J_{0}(x)\) given by Formula (4) of Section  satisfies the differential equation \(x y^{\prime \prime}+y^{\prime}+x y=0\). (This is called the Bessel equation of order zero.)

(b) Show that the Bessel function \(J_{1}(x)\) given by Formula (5) of Section  satisfies the differential equation \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\). (This is called the Bessel equation of order one.)

(c) Show that \(J_{0}^{\prime}(x)=J_{1}(x)\).

Equation  Transcription:

Text Transcription:

J_0(x)

xy”+y’+xy=0

J_1(x)

x^2y”+xy’+(x^2-1)y=0

J_0(x)=-J_1(x)

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