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Mathematic Methods HW Solutions 41

Mathematic Methods HW Solutions 41 - Chapter 8 41 1 3.13 x...

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Chapter 8 41 3.13 x = 1 2 e y + Ce - y 3.14 x = y 2 / 3 + Cy - 1 / 3 3.15 S = 1 2 × 10 7 bracketleftbig (1 + 3 t/ 10 4 ) + (1 + 3 t/ 10 4 ) - 1 / 3 bracketrightbig , where S = number of pounds of salt, and t is in hours. 3.16 I = Ae - Rt/L + V 0 ( R 2 + ω 2 L 2 ) - 1 ( R cos ωt + ωL sin ωt ) 3.17 I = Ae - t/ ( RC ) - V 0 ωC (sin ωt - ωRC cos ωt ) / (1 + ω 2 R 2 C 2 ) 3.18 RL circuit: I = Ae - Rt/L + V 0 ( R + iωL ) - 1 e iωt RC circuit: I = Ae - t/RC + iωV 0 C (1 + iωRC ) - 1 e iωt 3.19 N 2 = N 0 λte - λt 3.20 N 3 = c 1 e - λ 1 t + c 2 e - λ 2 t + c 3 e - λ 3 t , where c 1 = λ 1 λ 2 N 0 ( λ 2 - λ 1 )( λ 3 - λ 1 ) , c 2 = λ 1 λ 2 N 0 ( λ 1 - λ 2 )( λ 3 - λ 2 ) , c 3 = λ 1 λ 2 N 0 ( λ 1 - λ 3 )( λ 2 - λ 3 ) 3.21 N n = c 1 e - λ 1 t + c 2 e - λ 2 t + · · · , where c 1 = λ 1 λ 2 ...λ n - 1 N 0 ( λ 2 - λ 1 )( λ 3 - λ 1 ) ... ( λ n - λ 1 ) , c 2 = λ 1 λ 2 ...λ n - 1 N 0 ( λ 1 - λ 2 )( λ 3 - λ 2 ) ... ( λ n - λ 2 ) , etc . (all λ prime s different). 3.22 y = x + 1 + Ke x 3.23 x = 2 π - 1 / 2 e - y 2 integraldisplay y k e u 2 du 4.1 y 1 / 3 = x - 3 + Ce - x/ 3 4.2 y 1 / 2 = 1 3 x 5 / 2 + Cx - 1 / 2 4.3 y 3 = 1 / 3 + Cx - 3 4.4 x 2 e 3 y + e x - 1 3 y 3 = C 4.5 x 2 - y 2 + 2 x ( y + 1) = C 4.6 4 sin x cos y + 2 x - sin 2 x - 2 y - sin 2 y = C 4.7 x = y (ln x + C ) 4.8 y 2 = 2 Cx + C 2 4.9 y 2 = Ce - x 2 /y 2 4.10 xy = Ce x/y 4.11 tan 1 2 ( x + y ) = x + C 4.12 x sin( y/x ) = C 4.13 y 2 = - sin 2 x + C sin 4 x 4.14 y = - x - 2 + K ( x - 1) - 1 4.15 y = - x - 1 ln( C - x ) 4.16 y 2 = C ( C ± 2 x ) 4.17 3 x 2 y - y 3 = C 4.18 x 2 + ( y - k ) 2 = k 2 4.19 r = Ae - θ , r = Be θ 4.25 (a) y = C + x 2 x 2 ( C - x 2 ) (b) y = x ( C + e 4 x ) C - e 4 x (c) y = e x ( C - e 2 x ) C + e 2 x 5.1 y = Ae x + Be - 2 x 5.2 y = ( Ax + B ) e 2 x 5.3 y = Ae 3 ix + Be - 3 ix or other forms as in (5 . 24) 5.4 y = e - x ( Ae ix + Be - ix ) or equivalent forms (5 . 17), (5 . 18)
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