315_pdfsam_math 54 differential equation solutions odd

315_pdfsam_math 54 differential equation solutions odd -...

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Exercises 5.5 The frst equation yields z = x + 1 k ( mD 2 +2 k ) 2 [ x ]= 1 k 2 ( mD 2 k ) 2 1 [ x ] , (5.30) and so ( mD 2 k ) [ x ]+ ( mD 2 k ) ± 1 k 2 ( mD 2 k ) 2 1 [ x ] ² = ( mD 2 k ) 1 k 2 ( mD 2 k ) 2 2 [ x ]=0 . The characteristic equation For this homogeneous linear equation with constant coefficients is ( mr 2 k ) 1 k 2 ( mr 2 k ) 2 2 =0 , which splits onto two equations, mr 2 k r = ± i r 2 k m (5.31) and 1 k 2 ( mr 2 k ) 2 2=0 ( mr 2 k ) 2 2 k 2 ³ mr 2 k 2 k ´³ mr 2 k + 2 k ´ r = ± i s (2 2) k m ,r = ± i s (2 + 2) k m . (5.32) Solutions (5.31) and (5.32) give normal Frequences ω 1 = 1 2 π r 2 k m 2 = 1 2 π s (2 2) k m 3 = 1
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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