2220hw9sol

# 2220hw9sol - Math 2220 Section 3.3 Problem Set 9 Solutions...

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Math 2220 Problem Set 9 Solutions Spring 2010 Section 3.3 : 1. For the vector field F ( x,y ) = ( x,y ) do the following things: (a) Sketch the vector field. Solution. Along the x -axis we have F ( x, 0) = ( x, 0), pointing toward the origin. Along the y -axis we have F (0 ,y ) = (0 ,y ), pointing away from the origin. This gives a start in sketching the vector field, as shown in the figure. Looking on the lines y = x and y = x we get F ( x,x ) = ( x,x ) and F ( x, x ) = ( x,x ), which gives us a bit more, enough to predict what the rest will look like. (b) Determine the flow lines by solving the differential equations dx dt = x and dy dt = y . Your answer should be functions x ( t ) and y ( t ) each of which depends on an arbitrary constant. Solution. By inspection, dx dt = x has solutions x = Ae - t for A an arbitrary constant. More systematically, we can rewrite the equation dx dt = x as dx x = dt . Integrating both sides of this equation gives ln x = t + C , which implies x = e C e - t = Ae - t . In a similar way the equation dy dt = y gives y = Be t for B an arbitrary constant. (c) Check that the flow lines are hyperbolas by eliminating the variable t from your formulas in part (b). Solution. We have xy = Ae - t Be t = AB . This can be written as xy = C for C = AB . These curves are the hyperbolas y = C/x . (d) Find the values of the arbitrary constants that give the flow line c ( t ) such that c (0) = (2 , 1). Solution. For t = 0 the equation ( x,y ) = ( Ae - t ,Be t ) = (2 , 1) gives ( A,B ) = (2 , 1) so A = 2 and B = 1 and we have c ( t ) = (2 e - t ,e t ). 2. Sketch the vector field F ( x,y ) = ( x,x 2 ), determine the flow lines, and show the flow lines are parabolas. Solution. Notice that the formula ( x,x 2 ) is indepen- dent of y , so the vector field looks the same on each vertical line. Thus is suffices to sketch the vector field along the x -axis and then translate the result verti- cally up and down. This leads to the figure at the right.

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