prelim1_5 - Prelim 1 Solution Problem 5 x2 y HxL 2 x y HxL...

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Prelim 1 Solution Problem 5 x 2 y '' H x L + 2 xy ' H x L - 2 y H x L = 0 ; x > 0. ü a. y H x L = x r y ' H x L = rx r - 1 y '' H x L = r H r - 1 L x r - 2 Substitute these into the differential equation to obtain x 2 r H r - 1 L x r - 2 + 2 xrx r - 1 - 2 x r = 0 x r H r H r - 1 L + 2 r - 2 L = 0. x r is not zero for x > 0 so we are left with r H r - 1 L + 2 r - 2 = 0 r 2 + r - 2 = 0 H r - 1 L H r + 2 L = 0 r = 1, - 2. So solutions are y H x L = x y H x L = x - 2 . ü b. Set y H x L = c 1 x + c 2 x - 2 y
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