finalexam2930spr11 - Final Exam Math 2930 show work 5...

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Final Exam Math 2930 Spring 2011 show work, 5 problems, no calculators 1 a) (4 points) Write your name and section number on your exam booklet. b) (12 points) Find four solutions y 1 ( x ), y 2 ( x ), y 3 ( x ), and y 4 ( x ) to y = - xy 2 with initial values at x = 2: y 1 (2) = 0, y 2 (2) = 1 4 , y 3 (2) = 1 2 , and y 4 (2) = 1, Plot them all on one graph, showing the interval of x for which each is defined. c) (4 points) Write a clear definition for the sentence “The 0 solution is stable.” After defining it, determine whether it is true for the equation of 1b). 2) In this problem consider the heat equation u t = u rr + 1 r u r + 1 r 2 u θθ for a function u ( r, θ, t ) defined in the plane using polar coordinates and time. a) (8 points) Find a steady state solution u ( r, θ ) in the disk ( r < 4), hav- ing boundary values u (4 , θ ) = 2 cos( θ ) - cos(2 θ ) on the circle r = 4. Since the solution might be on your formula sheet, you must either derive your solu- tion by separation of variables, or verify explicitly that your solution meets all requirements.
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