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Unformatted text preview: MA 2930, March 30, 2011 Worksheet 9 1. Find the solution (if possible) of y 00 + y = 0 for the following sets of boundary values: (a) y (0) = 0 , y ( ) = 0 (b) y (0) = 0 , y ( ) = 0 (c) y (0) = 0 , y ( ) = 0 (d) y (0) = 0 , y ( ) = 0 How would you make sense of these results? (Hint: think graphically!) For which of these boundary values would the equation y 00 + y = cos2 x be solvable? The general solution of the differential equation is y ( x ) = c 1 cos x + c 2 sin x Now lets apply the boundary conditions: (a) y (0) = c 1 = 0 and y ( ) = c 1 = 0. So the solutions are y ( x ) = c 2 sin x . Graphically this makes sense because any sine with period 2 can satisfy the boundary conditions, but no cosine. Note that there are infinitely many solutions, illustrating that there is no uniqueness result for BVPs. (b) y (0) = c 1 = 0 and y ( ) = c 2 = 0. So the only solution is y ( x ) = 0. (c) y (0) = c 2 = 0 and y ( ) = c 1 = 0. So the only solution is again y ( x ) = 0. (d) y (0) = c 2 = 0 and y ( ) = c 2 = 0. So the solutions are y ( x ) = c 1 cos x . The general solution of the differential equation of y 00 + y = cos2 x is y ( x ) = c 1 cos x + c 2 sin x 1 / 3cos2 x Now lets apply the boundary conditions:...
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This note was uploaded on 06/10/2011 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).
 Spring '07
 TERRELL,R
 Differential Equations, Equations, Sets

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