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Unformatted text preview: 2 MA 36600 LECTURE NOTES: MONDAY, MARCH 30 (3) Y = Y (t) is a solution to the nonhomogeneous equation n ￿ j =0 Pn−j (t) Y (j ) = G(t). Then the general solution to the differential equation is y (t) = yc (t) + Y (t) where yc (t) = n ￿ Ci yi (t). i=1 For example, consider now the constant coefficient equation a0 Say that we can factor Z (r) = n ￿ j =0 dn y dn−1 y dy + a1 n−1 + · · · + an−1 + an y = G(t). n dt dt dt an−j rj = a0 · p ￿￿ k=1 ￿ r − rk ￿￿ ￿ q￿ ￿ sk ￿ ￿ ￿￿ ￿ sk · r − [λk + i µk ] r − [λk − i µk ] . real roots k=1 ￿￿ ￿￿ complex roots ￿ Then the general solution is y (t) = yc (t) + Y (t), where ￿s ￿ ￿￿ s ￿ ￿s ￿ ￿ p q k k k ￿￿ ￿ ￿ ￿ m−1 rk t m−1 λk t m−1 λk t yc (t) = Ckm t e+ Akm t e cos µk t + Bkm t e sin µk t . k=1 m=1 m=1 k=1 m=1 We discuss a couple of ways to compute Y = Y (t). Method of Undetermined Coefficients. Consider the nonhomogeneous differential equation dn y dn−1 y dy + a1 n−1 + · · · + an−1 + an y = G(t). n dt dt dt We outline a method to determine a particular solution Y = Y (t). #1. Express the function on the right-hand side as the sum of functions a0 G(t) = G1 (t) + G2 (t) + · · · + Gr (t) where each Gi (t) is the product of a polynomial, an exponential function, and a trigonometric function. That is, say that we can write di di ￿ ￿ Gi (t) = aij tj eαi t cos βi t + bij tj eαi t sin βi t j =0 j =0 for some constants aij , bij , αi , and βi . Note that Gi (t) involves a polynomial of degree di . #2. Make a guess that a solution Yi = Yi (t) of the nonhomogeneous equation n ￿ m=1 is in the form (m) Pn−m (t) Yi di +n ￿ Yi (t) = j =0 (t) = Gi (t) for di +n ￿ Aij tj eαi t cos βi t + j =0 i = 1, 2, . . . , r; Bij tj eαi t sin βi t for some constants Aij and Bij . Note that αi , and βi are the same as above, and that Yi (t) involves a polynomial of degree (di + n). #3. Recombine as the sum Y (t) = Y1 (t) + Y2 (t) + · · · + Yr (t); then Y = Y (t) is the desired solution to the nonhomogeneous equation a0 dn Y dn−1 Y dY + a1 n−1 + · · · + an−1 + an Y = G(t). n dt dt dt ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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