hw8 - y 1 and y 2 is a multiple of the other. In problems...

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Math 135A, Winter 2008 Homework #8 (due 2/29/2008) Routine problems: § 9.3: #11, #13, #23, #26, #10, #19, #21, #37; To hand in: (1) Show tht X k =0 sin k 2 k converges. Evaluate it exactly bu using the fact that sin k = = ( e ik ). Hint: It will be useful to recall that 1 a + ib = a a 2 + b 2 - i b a 2 + b 2 , ( a,b real) . (2) Let y 1 and y 2 be solutions of y 00 + py 0 + qy = 0, where p , q are continuous functions on the interval I = ( a,b ). Show that if there is a point in I where both y 1 and y 2 both vanish or where both have maxima or minima, then one of
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Unformatted text preview: y 1 and y 2 is a multiple of the other. In problems 35, you are given a dierential equation and one solution of it. Use reduction of order to nd the general solution. (3) x 2 y 00-x ( x + 2) y + ( x + 2) y = 0, y 1 ( x ) = x . (4) xy 00-( x + 2) y + 2 y = 0, y 1 ( x ) = e x . (5) x 2 y 00 + xy + ( x 2-1 4 ) y = 0, y 1 ( x ) = x-1 / 2 sin x . 1...
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