# Exam3past - y(0 = 1 using the method of Laplace transforms 4 M427K Exam 3 May 4 2006 Question 4[10 Points Solve the two-point boundary value

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Math 427K Exam 3 Name: UT EID: Instructions Please put your name and UT EID in the space provided. There are 6 questions each worth 10 points. You have 75 minutes to complete the test. Please write your working and solutions on the test paper. You may use the back of the pages. Calculators are not allowed. For Instructor’s Use Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Total

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M427K Exam 3 May 4, 2006 Question 1 [10 Points] Solve the initial value problem y 00 + 2 y 0 + y = e - t , y (0) = 1 , y 0 (0) = - 1 using the method of Laplace transforms. Hint: It is constant coeﬃcient second order linear so you know the terms that must appear in the solution. 2
M427K Exam 3 May 4, 2006 Question 2 1. [5 Points] Find the Laplace transform of the piecewise deﬁned function f ( t ) = 0 t < 4 t 4 t < 8 0 8 t 2. [5 Points] Find the inverse Laplace transform of s + 1 s 2 - 10 s + 29 . 3

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M427K Exam 3 May 4, 2006 Question 3 [10 Points] Solve the initial value problem y 00 + 3 y 0 + 2 y = δ ( t - 5) , y (0) = 0 ,

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Unformatted text preview: y (0) = 1 using the method of Laplace transforms. 4 M427K Exam 3 May 4, 2006 Question 4 [10 Points] Solve the two-point boundary value problem y 00 + λy = 0 , y (0) = y ( L ) = 0 by ﬁnding the eigenvalues and eigenfunctions. You may assume that there are no negative eignevalues. 5 M427K Exam 3 May 4, 2006 Question 5 [10 Points] Compute the Fourier cosine series for the function f : [0 , 1] → R given by f ( x ) = 1-x 2 . 6 M427K Exam 3 May 4, 2006 Question 6 [10 Points] Solve the homogeneous heat conduction equation for a bar of length 1 100 u xx = u t , < x < 1 , t > subject to the homogeneous boundary values u (0 , t ) = 0 , u (1 , t ) = 0 and the initial condition u ( x, 0) = sin(3 πx )-sin(7 πx ) . You do not need to ﬁnd the basic solutions, you may use the solutions found in class. Hint: No Fourier series is required. 7...
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## This note was uploaded on 04/30/2009 for the course M 58055 taught by Professor Windsor during the Spring '07 term at University of Texas at Austin.

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Exam3past - y(0 = 1 using the method of Laplace transforms 4 M427K Exam 3 May 4 2006 Question 4[10 Points Solve the two-point boundary value

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