/QuickPDFImd8996ec6 418 0 R /Type /StructElem x >> 1 209 0 R 210 0 R 213 0 R 214 0 R 215 0 R 216 0 R ] endobj >> /S /P endobj 289 0 obj /K [ 17 ] /Nums [ 0 57 0 R 1 107 0 R 2 160 0 R 3 218 0 R 4 279 0 R 5 331 0 R ] >> endobj These lecture notes are intended for the courses “Introduction to Mathematical Methods” and “Introduction to Mathematical Methods in Economics”. /P 54 0 R /S /P /P 54 0 R /K [ 17 ] << /S /P Annihilator Method. >> /Slide /Part << >> >> /P 173 0 R {\displaystyle \{2+i,2-i,ik,-ik\}} . /S /LBody >> >> /Pg 39 0 R ⁡ 55 0 obj /P 54 0 R /Pg 36 0 R << /S /LBody ( /K [ 6 ] 286 0 obj 172 0 obj /Pg 26 0 R /K [ 5 ] 256 0 obj /P 54 0 R /P 54 0 R << + endobj >> endobj /K [ 15 ] << /K [ 14 ] /P 271 0 R /K [ 29 ] /Pg 3 0 R endobj /Type /StructElem /Pg 39 0 R << {\displaystyle A(D)f(x)=0} /Type /StructElem /P 54 0 R k /P 281 0 R /S /P >> We work a wide variety of examples illustrating the many guidelines for making the 338 0 obj << /K [ 267 0 R ] ⁡ 194 0 obj /Type /StructElem = << + Keywords: ordinary differential equations; linear equations and systems; linear differential equations; complex exponential AMS Subject Classifications: 34A30; 97D40; 30-01 1. , and a suitable reassignment of the constants gives a simpler and more understandable form of the complementary solution, /S /P /Pg 3 0 R /Pg 39 0 R << /P 54 0 R << /P 54 0 R << /Pg 3 0 R /P 54 0 R >> endobj << << /S /P {\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}=\{e^{(2+i)x},e^{(2-i)x},e^{ikx},e^{-ikx}\}. /Type /StructElem /P 54 0 R /K [ 21 ] 1 /P 54 0 R /S /P /S /P 203 0 obj /Type /StructElem /Pg 39 0 R P >> Annihilator of eαt cosβt, cont’d In general, eαt cosβt and eαt sinβt are annihilated by (D −α)2 +β2 Example 4: What is the annihilator of f = ert? >> /P 238 0 R 1. /K [ 32 ] /Pg 41 0 R 335 0 obj /P 54 0 R /K [ 33 ] /P 54 0 R /P 54 0 R /K [ 23 ] /S /P /QuickPDFIm715354ce 419 0 R << /K [ 229 0 R ] >> /Type /StructElem /S /P − << 99 0 obj >> << {\displaystyle \sin(kx)} endobj >> /Pg 41 0 R 213 0 obj /Pg 39 0 R /K [ 22 ] 224 0 R 225 0 R 226 0 R 229 0 R 230 0 R 231 0 R 232 0 R 233 0 R 234 0 R 235 0 R 236 0 R /Type /StructElem endobj + /Pg 41 0 R /S /Figure /K [ 33 ] /S /LBody cos Example. /K [ 39 ] /P 54 0 R In the example b, we have already seen that, okay, D squared + 2D + 5, okay, annihilates both e to the -x cosine 2x and e to the -x sine 2x, right? >> ODEs: Using the annihilator method, find all solutions to the linear ODE y"-y = sin(2x). << /K 6 /Pg 41 0 R /K [ 29 ] endobj /P 54 0 R << /Type /StructElem If Lis a linear differential operator with constant coefficients and fis a sufficiently differentiable function such that [ ( )]=0. /Pg 26 0 R − >> endobj c /XObject << >> /S /P e ( /S /P >> /S /P endobj … /Type /StructElem /K [ 42 ] /P 54 0 R >> /K [ 16 ] {\displaystyle y_{p}={\frac {4k\cos(kx)+(5-k^{2})\sin(kx)}{k^{4}+6k^{2}+25}}} /S /P So we found that finally D squared + 2D + 5, cubed, is an annihilator of all these expression down here, okay. y Then the original inhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combination to satisfy the ODE. /K [ 271 0 R ] 101 0 obj 125 0 obj << 207 0 obj 140 0 obj /K [ 262 0 R ] /P 54 0 R 142 0 R 143 0 R 144 0 R 145 0 R 146 0 R 147 0 R 148 0 R 149 0 R 150 0 R 151 0 R 152 0 R 337 0 obj endobj endobj /ParentTreeNextKey 6 D , /K [ 4 ] << /K [ 39 ] c /S /P /P 54 0 R 104 0 obj /Pg 3 0 R /K [ 24 ] The DE to be solved has again the same limitations (constant coefficients and restrictions on the right side). << /P 54 0 R endobj (iii) The differential operator whose characteristic equation i! /K [ 41 ] endobj /P 54 0 R 70 0 R 71 0 R 72 0 R 73 0 R 74 0 R 75 0 R 76 0 R 77 0 R 78 0 R 79 0 R 80 0 R 81 0 R /Type /StructElem /Pg 26 0 R /K [ 0 ] /K [ 12 ] /Pg 39 0 R >> /P 261 0 R << 4 + /K [ 5 ] 131 0 obj 176 0 obj /Type /StructElem /S /P /Pg 39 0 R << 108 0 obj /S /L /K [ 19 ] 1 } endobj /K [ 47 ] /P 54 0 R /Pg 3 0 R /S /L /K [ 32 ] >> /Pg 26 0 R /Pg 3 0 R /Type /StructElem /K [ 34 ] endobj /S /P /S /P /Pg 26 0 R /F8 22 0 R z /K [ 282 0 R ] << /K [ 3 ] >> 2 >> k We demonstrate a successful example of in silico discovery of a novel annihilator, phenyl-substituted BTD, and present experimental validation via low temperature phosphorescence and the presence of upconverted blue light emission when coupled to a platinum octaethylporphyrin (PtOEP) sensitizer. /Metadata 376 0 R /K [ 7 ] << /Type /StructElem /S /P /K [ 59 ] endobj << /S /P >> >> /S /P << /P 123 0 R >> 119 0 obj /K [ 12 ] /Pg 26 0 R endobj /S /L /Type /StructElem c y 96 0 R 97 0 R 98 0 R 99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R 105 0 R ] << /K [ 38 ] /Pg 48 0 R /MediaBox [ 0 0 612 792 ] << D /S /P /K [ 38 ] endobj >> Answer: It is given by (D −r), since (D −r)f = 0. /P 54 0 R /Type /StructElem /S /P << /P 54 0 R >> 294 0 obj we give two examples; the first illustrates again the usefulness of complex exponentials. 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