It can be shown that J0 has infinitely many zeros for x > 0. In particular, the first threezeros are approximately 2.405, 5.520, and 8.653 (see Figure 5.7.1). Let j, j = 1, 2, 3, ... ,denote the zeros of J0; it follows thatJ0(jx) =1, x = 0,0, x = 1.Verify that y = J0(jx) satisfies the differential equationy +1xy + 2j y = 0, x > 0.Hence show that 10xJ0(ix)J0(jx) dx = 0 if i = j.This important property of J0(ix), which is known as the orthogonality property, is usefulin solving boundary value problems.Hint: Write the differential equation for J0(ix). Multiply it by xJ0(jx) and subtract itfrom xJ0(ix) times the differential equation for J0(jx). Then integrate from 0 to 1.

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