practice2 - a, b ). 4. Using the method of eigenfunction...

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Practice Exam 2: MAP 4305 * 1. Find the positive eigenvalues and eigenfunctions of the following eigenvalue problem: y ′′ + λy = 0 , y (0) + y (0) = y ( π ) = 0 . 2. Find conditions on f so that the following non-homogeneous BV problem has a solution: y ′′ - y + 3 y = f, y (0) = y ( π ) = 0 3. Consider the linear operator L [ y ] = y (4) , de±ned on the set of functions having continuous derivatives up to order 4, that satisfy the following periodic boundary conditions: y ( i ) ( a ) = y ( i ) ( b ) , i = 0 , 1 , 2 , 3 . Show that L is self-adjoint with respect to the usual inner product for functions on (
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Unformatted text preview: a, b ). 4. Using the method of eigenfunction expansion, solve the following non-homogeneous BV problem: y + 9 y = 1 + cos( x ) , y (0) = y ( ) = 0 , given that the eigenvalues and eigenfuctions of y + y = 0 , y (0) = y ( ) = 0 are n = n 2 , n = 0 , 1 , . . . , and n ( x ) = c n cos( nx ) where the c n are arbitrary nonzero constants. * Instructor: Patrick De Leenheer....
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