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Midterm2 - MAE105 Mid-term 2(open book closed notes Name...

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MAE105 Mid-term 2 (open book, closed notes) Name:_________________________ Time: 3:35 to 4:45pm Date: May 22, 2007 Problem 1 (a) (1 Point) Find a general expression x = x ( t ), for the characteristics of the following PDE: u t - x cos t u x = - u sin t . [Note that your expression must include a constant of integration, say, x (0) = x 0 .] (b) (1 Point) In the x , t -plane, t > 0, sketch a typical characteristic curve. (c) (1 Point) Find the general solution of this PDE. (d) (1 Point) Specialize the solution in (b) such that at t = 0, we have u ( x 0 ,0) = u 0 = ( x 2 0 + 1). Problem 2 Consider the wav e equation 2 u t 2 - 2 u x 2 = 0 (1) in an infinite domain, - < x < , t > 0, with the initial conditions u ( x , 0) = f ( x ) = x 1 - x for 0 < x < 1 / 2 for 1 / 2 < x < 1 , u t ( x , 0) = g ( x ) = sin( π x ) , 0 < x < 1 , with both f ( x ) and g ( x ) being zero for all other values of x . (a) (1 Point) Draw the ( x , t )-plane, and below the x -axis, draw the initial conditions in two graphs, as discussed in the class and in your book. (b) (1.5 Points) In the x , t -plane, draw the characteristics that pass through the following points: [ x = 0, t =
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