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Unformatted text preview: Math 425 Dr. DeTurck Hints and Solutions to Practice Midterm 1 February 2010 1. Suppose f is a function of one variable that has a continuous second derivative. Show that for any constants a and b , the function u ( x,y ) = f ( ax + by ) is a solution of the PDE u xx u yy- u 2 xy = 0 . This is an exercise in using the chain rule. For instance, u x ( x,y ) = af ( ax + by ), u xx = a 2 f 00 ( ax + by ), etc., so eventually: u xx u yy- u 2 xy = a 2 b 2 ( f 00 ( ax + by )) 2- ( ab ) 2 ( f 00 ( ax + by )) 2 = 0 . 2. Give an example that shows why solutions of the wave equation u tt = u xx do not necessarily satisfy the maximum principle (i.e., give an example of an explicit solution of the equation for which the maximum principle does not hold). For this, we need a solution to the wave equation for x (0 ,L ) and for t (0 ,T ) for which the maximum occurs in the interior of the rectangle. For instance, the function u ( x,t ) = sin x sin t satisfies the wave equation, but the maximum of u = 1 occurs when x = t = / 2, in the interior of the rectangle [0 , ] [0 , ] (where u = 0 identically on the boundary of the rectangle)....
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