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# undetermined_coef_1 - Undetermined coe hw 3.5 5 9 17(graded...

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Undetermined coeff. hw 3.5: 5, 9, 17(graded), 22a, 23a October 4, 2010 (5) Find the general solution y + 9 y = t 2 e 3 t + 6 (1) First solve for the homogeneous solution y h . The characteristic polynomial is r 2 + 9 = 0 and has roots r = ± 3 i y h = C 1 cos 3 t + C 2 sin 3 t (2) Next we assume a form of the particular solution based as a linear combination of the non-homogenous terms and all their derivatives. y p i = e 3 t ( A t 2 + B t + C ) + D (3) where the [] group terms that correspond to a single term in our non-homogeneous function. Since there are no terms in y p i that appear in y h we can finalize the appropriate form of our par- ticular solution y p and conclude y p = y p i . Differentiating y p twice gives, y p = e 3 t ( A t 2 + B t + C ) + D (4) y p = A ( 2 e 3 t + 12 t e 3 t + 9 t 2 e 3 t ) + B ( 6 e 3 t + 9 t e 3 t ) + 9 C e 3 t (5) Substituting (4) and (5) into (2) and collecting like terms of t we find, t 2 e 3 t 18 A = 1 A = 1 / 18 (6) t e 3 t 12 A + 18 B = 0 B = - 1 / 27 (7) e 3 t 2 A + 6 B + 18 C = 0 C = 1 / 162 (8) Constants 9 D = 6 D = 2 / 3 (9) (10) y p = 1 / 18 t 2 e 3 t - 1 / 27 t e 3 t + 1 / 162 e

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undetermined_coef_1 - Undetermined coe hw 3.5 5 9 17(graded...

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