# final - 18.034 FINAL EXAMINATION Formulas of the Laplace...

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18.034 FINAL EXAMINATION May 22, 2007 Formulas of the Laplace transform (1) L [ f ( t )]( s ) = F ( s ) = f ( t ) e st dt for s > 0 . 0 1 (2) L [1] = . s 1 (3) L [ e at ] = . s a n ! (4) L [ t n ] = . s n +1 s (5) L [cos( ωt )] = . s 2 + ω 2 ω (6) L [sin( ωt )] = . s 2 + ω 2 e as (7) L [ u a ( t )] = . s (8) L [ δ a ( t )] = e as . Fourier series a 0 πx πx 2 πx 2 πx f ( x ) = + a 1 cos + b 1 sin + a 2 cos + b 2 sin + 2 l l l l ··· , where l l 1 nπx 1 nπx a n = f ( x ) cos dx, b n = f ( x ) sin dx. l l l l l l 1

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18.034 Final Exam Name: 1. Consider the ﬁrst-order DE y = y 3 y . (a) (5 pts) Find all critical points of the DE, that is, ﬁnd all solutions for which y = 0 . (b) (10 pts) Use the method of separation of variables, ﬁnd a general solution of the DE. 2
Final Exam Name: (c) (15 pts) Study the stability property at each of the critical points of part (a). 3

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## This note was uploaded on 07/28/2008 for the course MATH 18.034 taught by Professor Hur during the Spring '07 term at MIT.

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final - 18.034 FINAL EXAMINATION Formulas of the Laplace...

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