Scott Armstrong math 54 final 2008

# Scott Armstrong math 54 final 2008 - u t ( x,t ) = 2 u x 2...

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Math 54 - Sample Final Exam August 15, 2008, 08:00-10:00 Name: This is a closed book, closed notes exam. Calculators are not allowed. You have two hours to complete the exam. To receive full credit, write legibly, show your work and write proofs in complete sentences. If you need more space, use the back of the page of the problem on which you are working. Problem Points Your Score 1 20 2 20 3 20 4 20 5 20 6 20 Total 120

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1. Find the solution of the initial-value problem ( 2 y 00 + 9 y 0 + 10 y = 8 sin t + 9 cos t y (0) = 4 , y 0 (0) = - 6 .
2. Find the solution of the initial-value problem x 0 ( t ) = 2 1 2 1 3 1 2 1 2 x ( t ) , x (0) = 1 1 1 .

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3. Find a general solution for the ODE 4 y (5) + 8 y (4) + 5 y 000 + 4 y 00 + 8 y 0 + 5 y = 0 . (Hint: the function y ( t ) = e - t sin( t 2 ) is one solution.)
4. Find the solution of the heat ﬂow problem

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Unformatted text preview: u t ( x,t ) = 2 u x 2 (0 &lt; x &lt; , t &gt; 0) u (0 ,t ) = u ( ,t ) = 0 ( t &gt; 0) u ( x, 0) = e-x (0 &lt; x &lt; ) 5. Find the formal solution of the problem 2 u x 2 ( x,y ) + 2 u y 2 ( x,y ) = 0 (0 &lt; x &lt; , &lt; y &lt; 1) u (0 ,y ) = u ( ,y ) = 0 (0 &lt; y &lt; 1) u y ( x, 0) = 0 , u ( x, 1) = f ( x ) (0 &lt; x &lt; ) 6. Let A be a square matrix, and let p be the characteristic polynomial of A . (a) If q is a polynomial such that q ( A ) = 0, show that q ( ) = 0 for each eigenvalue of A . (b) Show that if A is diagonalizable then p ( A ) = 0....
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## This note was uploaded on 01/12/2010 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at University of California, Berkeley.

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Scott Armstrong math 54 final 2008 - u t ( x,t ) = 2 u x 2...

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