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108_11_2009_second_exam

# 108_11_2009_second_exam - Second Midterm Examination MAT...

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Second Midterm Examination, MAT 108, Differential Equations. Last Name First Name Student ID Number Remarks: The test contains 5 pages, including the cover sheet. Do not de-staple the test. The use of books, notes, and calculators is not allowed. You can use scratch paper during the exam. However, you cannot submit any work on your scratch paper with the exam. Use both sides of the sheets.

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Exercise I. Let f be a 2 p - periodic function such that f ( x ) = a 0 + X n =1 a n cos p x + b n sin p x , (1) with a 0 , a i and b i , i = 1 , · · · , n , and p are real numbers. Prove the Parseval’s identity 1 2 p Z p - p [ f ( x )] 2 dx = a 2 0 + 1 2 X n =1 ( a 2 n + b 2 n ) . (2) Show that the Fourier series decomposition of the 2 π -periodic function f ( x ) = x/ 2 for - π < x < π is given by X n =1 ( - 1) n +1 n sin( nx ) , - π < x < π. (3) Use Parseval’s identity to deduce that n =1 1 n 2 = π 2 6 , k =1 1 (2 k ) 2 = π 2 24 and k =0 1 (2 k +1) 2 = π 2 8 .
Exercise II. Find the Laplace transform of f ( t ) = 1 if 0 t π , f ( t ) = - 1 if

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