09fall2 - MTH 108 Fall 09-10 Exam 2 Show all work Justify...

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MTH 108 Fall 09-10 Exam 2, Show all work. Justify procedures and calculations except when they are obvious as, for example, when you integrate both sides of an equation. When doing so, use complete, grammatically correct sentences. No books, notes, calculators. 1. (a) Find a solution of the third order ODE in the half-line 0 < x < , u ±±± + 8 u = 0 , which satisfies the conditions that u (0) = 5 and that u ( x ) decays to zero as x tends to infinity. Do you think the solution is unique? (b) Find the general solution of the third order ODE u ±±± + 4 u ± = 0 2. Let the real valued functions f ( x ) and g ( x ) be defined, in 0 < x < 1, by f ( x ) = a 1 φ 1 ( x ) + a 2 φ 2 ( x ) + a 3 φ 3 ( x ) + ··· , g ( x ) = b 1 φ 1 ( x ) + b 2 φ 2 ( x ) + b 3 φ 3 ( x ) + ··· , where φ 1 , φ 2 , φ 3 , ··· is a given orthonormal basis of real valued functions in the interval 0 < x < 1, with inner (dot) product ( f, g ) = Z 1 0 f ( x ) g ( x ) dx. (a) Prove that
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This note was uploaded on 01/17/2010 for the course MATH 108 taught by Professor Trangenstein during the Fall '07 term at Duke.

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09fall2 - MTH 108 Fall 09-10 Exam 2 Show all work Justify...

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