practice4 - 3 Find the adjoint problem of x 2 y 00-xy 3 y =...

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Practice Exam 4: MAP 4305 * 1. It is known that Hermite’s equation y - 2 xy + 2 ny = 0 , where n is a nonnegative integer has polynomial solutions of degree n . Denote these solutions by H n ( x ). There is a generating function for these polynomials: e 2 tx - t 2 = n =0 H n ( x ) t n n ! Using the generating function, determine H 0 ( x ), H 1 ( x ) and H 2 ( x ). Let n = 2 and find H 2 ( x ) by solving Hermite’s equation directly (notice that x = 0 is an ordinary point). 2. Find (graphically and approximately) the positive eigenvalues and eigenfunctions of the following eigenvalue problem: y + λy = 0 , y (0) = y ( π ) - y ( π ) = 0 . Provide an open interval of the form (0 , c ) for some to be determined c > 0 which contains λ 1 , the smallest positive eigenvalue.
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Unformatted text preview: 3. Find the adjoint problem of x 2 y 00-xy + 3 y = 0 , y (1) = y (2) = 0 . Is it self-adjoint? 4. Consider the linear operator L [ y ] = y (4) , defined on the set of functions having continuous derivatives up to order 4, that satisfy the following periodic boundary conditions: y ( i ) (0) = y ( i ) (1) , i = 0 , 1 , 2 , 3 . Show that L is self-adjoint, ie show that ( L [ y 1 ] , y 2 ) = ( y 1 , L [ y 2 ]) for all y 1 and y 2 in the domain of L where ( y 1 , y 2 ) denotes the usual inner product of the functions y 1 and y 2 . * Instructor: Patrick De Leenheer. 1...
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