u_x_plus_3u_yeq0

u_x_plus_3u_yeq0 - x y one computes y-3 x to determine C...

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Math 260, Spring 2012 Jerry L. Kazdan u x + 3 u y = 0 This example is similar to Problem Set 7 # 3a). Example : Find a function u ( x, y ) that satisfies u x + 3 u y = 0 with u (0 , y ) = 1 + e 2 y . Solution :The differential equation can be written u · V = 0 where V = (1 , 3). It means that at every point the directional derivative in the direction of V is 0 so u ( x, y ) is constant along these parallel straight lines, which have the form y = 3 x + C . Given a point (
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Unformatted text preview: x, y ) one computes y-3 x to determine C , that is, which line you are on. Thus the solution u ( x, y ) = h ( y-3 x ) for some as yet unknown function h ( s ). Now we use the initial condition u (0 , y ) = 1 + e 2 y . It gives us 1 + e 2 y = u (0 , y ) = h ( y ) . Consequently u ( x, y ) = 1 + e 2( y-3 x ) . [Last revised: February 28, 2012] 1...
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