18.034, Honors Differential Equations Page 1 of 2 Prof. Jason Starr 18.034 ; Feb 6, 2004 Lecture 2 1. Set-up model for a mixing problem Rate of mass of chemical in =(conc. in)×(rate of flow of liquid)= a.c(t) Rate of mass out = (conc. out) ×(rate of flow) = ( )vtyq ⋅. So ()yvatcay −⋅=' where a, 0>V are constants 2. Discussed method of integrating factors: (a) Guess there exists ( )tu s.t. ( ) ( ) ( )ytptuytu +' equals ( )[ ]'ytu . (b) This leads to assorted separable equations, ( )utpu =' which has a solution ()()tppdteetU ==∫, which is evacuated. (c) Define ()yextp= , ()xeytp= . Then ( ) ( )tqytpy =+' iff ()()tqextp=' . Moreover, choosing ( ) ( )00,00 yyP == iff ()00 yx = . So have existence/ uniqueness of original IVP iff existence/ uniqueness of IVP ( )( ) ( )00,' yxtqextp==. But this follows from F.T. of calculus. (d) Conclusion: If () ()tqtp , are defined and cts. on ( )ba,⊂ IR , then there exists a solution ()ty of () ( )tqytpy =+' defined on all of ( )ba, , the solution is unique, and it has the form. where ( ) ( )tptp =' ( )00 =p 3. Used this method to solve the mixing problem: () ()tvttvtveydsscaeetyααα−−+=∫00 (a) If ()ctc = is constant, get () ( )tveycVcVtyα−−−=0, i.e. ()()()tveycVtycVα−−=−0. ()tcayvay .'=+( ) ( )tqytpy =+'()() ()(),)(00tptsptpeydssqeety−−+=∫V=volume, a=rate of flow of solutionC(t)= concentration in y(t)= mass of chemical in tank at time t18.034, Honors Differential Equations Page 2 of 2 Prof. Jason Starr So equil. solution is cVy = (which makes physical sense), and the “half-life”, to come with 21 of equilibrium from initial value is aV=τln(2) (increasing V or decreasing a increases the half-life). (b) Consider the case that () ( )tAtcωsin⋅= . Let to integral ()dssaAetsvωαsin0∫. Set va=λ, get ()()λωφφωωλλ=−+tan,sin22teaAt . Didn’t have time to really analyze the solution. () ()tbetaAtyλφωωλ−+−+= sin22 for some b 4. Particular solution method. To find the general solution of ( )()tqytpy =+', (i) Find general solution of undriven/ homo system ( )00'0=+ ytpy. (ii) Find a particular solution py of original equation. (iii) General solution is pyy
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