02c lin ode cnst coef - Applied Mathematics 105b Feb 2008 ...

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Applied Mathematics 105bFeb 2008 (revision of Feb 1997 version)J. R. Rice Notes on linear ode's with constant coefficients(see also Greenberg, 1998, Sect. 3.4 & 3.7) Linear operatorLwith constant coefficients: Ly±dnydxn+a1dn²1 ydx n²1+...+an²1dy dx +an ywherea1,a2, ..,an–1,an are all constant Operation on exponential(r here corresponds to±of Greenberg, 1998): L(erx)=Pn (r)erx , where Pn(r)=rn+a1rn±1 +...+an±1r+an=(r±r1)(r±r2)...(r±rn±1)(r±rn ) Observations:(1)For any root, say,r=r1,Pn(r1 )=0; (2)Ifr1 is adoubleroot, thenPn(r1)=dPn(r1 ) /dr=0; (3)Ifr1is atripleroot, thenPn(r1)=dPn(r1) /dr=d2Pn(r1 ) /dr2 =0 ;etc. Now regarderx as a function of thetwovariablesrandx, so that we think ofLas (partial derivative notation)L=±n±xn+a1±n²1 ±x n²1 +...+an²1± ±x +an , and observe that L±±r(erx)=± ±r L(erx )-- exchange order of partial differentiation.Evaluating both sides, L[±±r(erx)]=L[xerx]=±±r[L(erx)]=±±r[Pn(r)erx]=[dPn (r) dr+xPn (r)]erx ,
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