set4_problems

set4_problems - MA 52700 Peter Scheiblechner Midterm Exam 2...

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Unformatted text preview: MA 52700, 07/26/2010 Peter Scheiblechner Midterm Exam 2 Your name: Your user ID (Purdue-email): 1 1. Mark the following functions, whose Laplace transform exists due to the existence theorem, with “Y”, and the others with “N”. f ( t ) = 1 1 − t f ( t ) = e e t f ( t ) = 1 − t 2 1 − t f ( t ) = e sin t f ( t ) = ln( t − 1) 2 Solution. N,N,Y,Y,N. 2 2. The inverse Laplace transform of 1 s 4 + π 2 s 2 is A. 1 π 2 ( t 2 − sin πt ) B. 1 π 2 ( t + 1 π sin πt ) C. 1 π 2 ( t + sin πt ) D. 1 π 2 ( t − 1 π sin πt ) E. 1 π 2 ( t 2 + 1 π sin πt ) Solution. D. 3 3. Solve the initial value problem y ′′ + 5 y ′ + 6 y = δ ( t − π 2 ) , y (0) = y ′ (0) = 0 using the Laplace transform. Solution. Writing Y := L ( y ) and applying the Laplace transform to the equation, we get s 2 Y + 5 sY + 6 Y = e − π 2 s ⇐⇒ Y = e − π 2 s s 2 + 5 s + 6 . Now the partial fraction decomposition is 1 s 2 + 5 s + 6 = 1 ( s + 2)( s + 3) = 1 s + 2 − 1 s + 3 , hence the inverse Laplace transform of Y = e − π 2 s s + 2 − e − π 2 s s + 3 is y = L − 1 ( Y ) = u ( t − π/ 2) e − 2( t − π 2 ) − u ( t − π/ 2) e − 3( t − π 2 ) ....
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This note was uploaded on 04/03/2012 for the course MA 527 taught by Professor Weitsman during the Spring '08 term at Purdue.

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set4_problems - MA 52700 Peter Scheiblechner Midterm Exam 2...

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