# hw2(1) - y ′′ y = sin 2 x y(0 = y(2 π y ′(0 = y...

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Homework assignment 2 * Due date: March 23. 2007. 1. Determine all real eigenvalues and corresponding eigenfunctions of y ′′ + λy = 0 , y (0) = 0 , y ( π ) + y ( π ) = 0 . 2. Consider the linear operator L [ y ] = y (4) , deFned on the set of functions having continuous derivatives up to order 4, that satisfy the following boundary conditions: y ( a ) = y ( a ) = y ( b ) = y ( b ) = 0 . Show that L is self-adjoint, ie that for all y 1 and y 2 in the domain of L , we have that: ( y 1 , L [ y 2 ]) = ( L [ y 1 ] , y 2 ) . Use this to prove that eigenfunctions of the problem: L [ y ] + λy = 0 , y ( a ) = y ( a ) = y ( b ) = y ( b ) = 0 corresponding to distinct eigenvalues are orthogonal, ie that ( y 1 , y 2 ) = 0 , whenever y 1 and y 2 are eigenfunctions corresponding to distinct eigenvalues. 3. Consider the following boundary value problem:
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Unformatted text preview: y ′′ + y = sin 2 x, y (0) = y (2 π ) , y ′ (0) = y ′ (2 π ) . • Using the ±redholm alternative, determine whether or not this problem has solutions. • If there are solutions, determine them. 4. Consider the following nonhomogeneous boundary value problem: y ′′ + y = f ( x ) , y (0) = y (1) = 0 , where f : [0 , 1] → R is given by f ( x ) = x (1-x ) . Determine a solution of this problem, written as a formal series expansion using an orthonor-mal system of eigenfunctions of the associated homogeneous problem. * MAP 4305; Instructor: Patrick De Leenheer. 1...
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