Chem Differential Eq HW Solutions Fall 2011 157

Chem Differential Eq HW Solutions Fall 2011 157 - Section...

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Section 12.5 Analytic Functions 157 The frst equation oF the Cauchy-Riemann equations tells us that u x = v y .So v y ( x, y )= e x cos y. Integrating with respect to y (treating x as a constant), we fnd v ( x, y e x sin y + c ( x ) , where the constant oF integration c ( x ) is a Function oF x . The second equation oF the Cauchy-Riemann equations tells us that u y = - v x v x ( x, y e x sin y + c ± ( x ); e x sin y + c ± ( x e x sin y ; c ± ( x )=0 ; c ( x C. Hence, v ( x, y e x sin y + C, which matches the previous Formula up to a additive constant. 37. (a) The level curves are given by u ( x, y y x 2 + y 2 = 1 2 C , where, For convenience, we have used 1 / (2 C ) instead oF the usual C For our arbitrary constant. The equation becomes x 2 + y 2 - 2 Cy =0 or x 2 +( y - C ) 2 = C 2 . Thus the level curves are circles centered at (0 ,C ) with radius C . (b) By Exercise 35, a harmonic conjugate oF u ( x, y )is v ( x, y x x 2 + y 2 . Thus the orthogonal curves to the Family oF curves in (a) are given by the level curves oF v ,or v ( x, y x x 2 + y 2 = 1 2 C or ( x - C ) 2 + y = C 2 . Thus the level curves are circles centered at (
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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