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automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Differential Operators, & Second Order Linear Differential Equations :[font = subtitle; inactive; preserveAspect] Steve Dunbar Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln, NE 68588-0323 sdunbar@mathlab01.unl.edu Fall Semester 1993 :[font = subsubtitle; inactive; preserveAspect] Covers Nagle & Saff, Section 4.2, 4.6 :[font = section; inactive; preserveAspect; startGroup] Experiments with Differential Operators :[font = subsection; inactive; preserveAspect; startGroup] Define some Linear Constant Coefficient Differential Operators :[font = text; inactive; preserveAspect] L1[y] = y'' - 2y' + y ;[s] 3:0,0;1,1;2,2;22,-1; 3:1,11,8,Times,0,12,0,0,0;1,11,8,Times,64,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] L1[f_] := D[f,{x,2}] - 2 D[f,x] + f :[font = text; inactive; preserveAspect] L2[y] = y''+4 y ;[s] 3:0,0;1,1;2,2;15,-1; 3:1,11,8,Times,0,12,0,0,0;1,11,8,Times,64,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect] L2[f_] := D[f,{x,2}] + 4 f :[font = text; inactive; preserveAspect] L3[y] = 2y'' + 9 y' + 9 y ;[s] 3:0,0;1,1;2,2;25,-1; 3:1,11,8,Times,0,12,0,0,0;1,11,8,Times,64,12,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; preserveAspect; endGroup] L3[f_] := 2 D[f,{x,2}] + 9 D[f,x] + 9 f :[font = subsection; inactive; preserveAspect; startGroup] Try some functions into these differential operators :[font = subsubsection; inactive; preserveAspect; startGroup] Experiments with L1 :[font = input; preserveAspect] L1[ x^3] :[font = input; preserveAspect] L1[ Sin[b x]] :[font = input; preserveAspect] L1[Cos[b x]] :[font = input; preserveAspect] L1[ Exp[a x]] :[font = input; preserveAspect] L1[ Exp[a x] Cos[b x] ] :[font = input; preserveAspect; endGroup] Simplify[%] :[font = subsubsection; inactive; preserveAspect; startGroup] Experinents with L2 :[font = input; preserveAspect] L2[ x^3] :[font = input; preserveAspect] L2[ Cos[b x]] :[font = input; preserveAspect] Simplify[%] :[font = input; preserveAspect] Simplify[ L2[ Sin[b x]]] :[font = input; preserveAspect; endGroup] Simplify[ L2[ Exp[ a x] Cos[ b x]]] :[font = subsubsection; inactive; preserveAspect; startGroup] Experiments with L3 :[font = input; preserveAspect] L3[ x^3] :[font = input; preserveAspect] Simplify[ L3[ Exp[ a x]]] :[font = input; preserveAspect; endGroup] Simplify[ L3[ Exp[ a x] Cos[ b x]]] :[font = subsubsection; inactive; preserveAspect; startGroup] Question to answer: :[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"; endGroup; endGroup; endGroup] What conclusion can you make about the image of an exponential function, a sinusoidal function, a polynomial, or a product of these under a linear constant coefficient differential operator? :[font = section; inactive; preserveAspect; startGroup] Experiments with Solutions of Second Order Linear Differential Equations :[font = subsection; inactive; preserveAspect; startGroup] Example of solution and plotting :[font = text; inactive; preserveAspect] Use the Mathematica command NDSolve to numerically solve second order differnetial equations. 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Perform :[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"] (Borelli and Coleman, Experiment 4.2.3 , page 122) Consider the IVP y'' - y' + b*y == 0, y(0) == 1, y'(0) == -1. Graph y(x) for several values of b, (positive, negative, and zero). Are there any values of b for which the solution is oscillatory? What do you guess is the general form of the solutions? :[font = text; inactive; preserveAspect; plain; bold; fontName = "Times"] (Borelli and Coleman, Experiment 4.2.4, page 122) Consider the IVP y'' - a*y' - y == 0, y(0) == 1, y'(0) == -1. Graph y(x) for several values of a, (positive, negative, and zero). Are there any values of a for which the solution is oscillatory? What do you guess is the general form of the solutions? :[font = text; inactive; preserveAspect] Extra Credit:: We suggested that the leading coefficient in a second order equation can't be 0. What happens to solutions if we try an equation with a leading coefficient nearly 0? :[font = input; preserveAspect; endGroup; endGroup; endGroup] Consider the IVP epsilon y'' + y' + y == 0, y(0) == 1, y'(0) == -1. Graph y(x) for several values of epsilon, (positive, negative, and zero). Are there any values of epsilon for which the solution is oscillatory? Does Mathematica make any complaints, or give any warning messages? ;[s] 3:0,0;231,1;242,2;294,-1; 3:1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0; ^*)