02c_lin_ode_cnst_coef

02c_lin_ode_cnst_coef - Applied Mathematics 105b Feb 2008...

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Applied Mathematics 105b Feb 2008 (revision of Feb 1997 version) J. R. Rice Notes on linear ode's with constant coefficients Linear operator L with constant coefficients : Ly ± d n y dx n + a 1 d n ² 1 y dx n ² 1 + ... + a n ² 1 dy dx + a n y where a 1 , a 2 , . ., a n–1 , a n are all constant Operation on exponential (r here corresponds to ± of Greenberg, 1998) : L ( e rx ) = P n ( r ) e rx , where P n ( r ) = r n + a 1 r n ± 1 + ... + a n ± 1 r + a n = ( r ± r 1 )( r ± r 2 )...( r ± r n ± 1 )( r ± r n ) Observations: (1) For any root, say, r = r 1 , P n ( r 1 ) = 0 ; (2) If r 1 is a double root, then P n ( r 1 ) = dP n ( r 1 )/ dr = 0 ; (3) If r 1 is a triple root, then P n ( r 1 ) = dP n ( r 1 )/ dr = d 2 P n ( r 1 )/ dr 2 = 0 ; etc. Now regard e rx as a function of the two variables r and x , so that we think of L as (partial derivative notation) L = n x n + a 1 n ² 1 x n ² 1 + ... + a n ² 1 x + a n , and observe that
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