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Chem Differential Eq HW Solutions Fall 2011 71

# Chem Differential Eq HW Solutions Fall 2011 71 - Section...

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Section 4.7 Bessel’s Equation and Bessel Functions 71 Using Exercise 22 and (1), you can also write this general solution in the form y = c 1 x ± sin x x - cos x ² + c 2 x ³ - cos x x - sin x ´ = c 1 [sin x - x cos x ]+ c 2 [ - cos x - x sin x ] . In particular, two linearly independent solution are y 1 = sin x - x cos x and y 2 = cos x + x sin x. This can be veriFed directly by using the diﬀerential equation (try it!). 17. We have y = x - p u, y ± = - px - p - 1 u + x - p u ± , y ±± = p ( p +1) x - p - 2 u +2( - p ) x - p - 1 u ± + x - p u ±± , xy ±± +(1+2 p ) y ± + xy = x µ p ( p x - p - 2 u - 2 px - p - 1 u ± + x - p u ±± +(1 + 2 p ) µ - px - p - 1 u + x - p u ± + xx - p u = x - p - 1 µ x 2 u ±± +[ - 2 px p ) x ] u ± +[ p ( p - (1+2 p ) p + x 2 ] u = x - p - 1 µ x 2 u ±± + xu ± +( x 2 - p 2 ) u . Thus, by letting y = x - p u , we transform the equation xy ±± p ) y ± + xy =0 into the equation x - p - 1 µ x 2 u ±± + xu ± x 2 - p 2 ) u , which, for x> 0, is equivalent to x 2 u ±± + xu ± x 2 - p 2 ) u , a Bessel equation of ordr
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