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Unformatted text preview: x 2 + 4 y 2 = c . (c) Taking logarithm of both sides of the equation, we obtain ln y = kx ln y x = k, and so F ( x, y ) = (ln y ) /x , F/x = (ln y ) /x 2 , F/y = 1 / ( xy ). The equation (2.17) becomes 1 xy dx ln y x 2 dy = 0 1 xy dx = ln y x 2 dy. Separating variables and integrating, we obtain x dx = y ln y dy Z x dx = Z y ln y dy x 2 2 = y 2 2 ln y + Z y 2 2 1 y dy = y 2 2 ln y + y 2 4 + c 1 x 2 2 + y 2 2 ln y y 2 4 = c 1 2 x 2 + 2 y 2 ln y y 2 = c, where c := 4 c 1 , and we have used integration by parts to nd R y ln y dy . 71...
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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