Differential Equations Lecture Work Solutions 147

Differential - Substitute in and compare coecients of cos w t Bn w 2 Cn w c2 n2 Bn = An Compare coecients of sin w t Cn w 2 Bn w Cn = 0 Bn = Cn(1 w

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Substitute in (*) and compare coefficients of cos wt B n w 2 + βC n w + c 2 n 2 B n = A n Compare coefficients of sin wt C n w 2 B n w + C n =0 B n = C n (1 w 2 ) w C n (1 w 2 ) w ( c 2 n 2 w 2 )+ βC n w = A n C n βw + c 2 n 2 w 2 w (1 w 2 ) | {z } = D n = A n C n = A n D n B n = A n (1 w 2 ) D n w u P n = A n D n (1 w 2 ) w cos wt + A n D n sin wt where D n = βw + c 2 n 2 w 2 w (1 w 2 ) A n = R π 0 sin nxdx R π 0 sin 2 nx dx Therefore the general solution of the inhomogeneous is u n = c 1 cos 4 c 2 n 2 β 2 2 t + c 2 sin 4 c 2 n 2 β 2 2 t ± e ( β/ 2) t ) + A n D n 1 w 2 w cos wt + A n D n sin wt (**) u ( x, t )= X n
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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