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### AMath350.A2.Sol

Course: AMATH 350, Fall 2011
School: Waterloo
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Word Count: 222

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y = 3x cos2 (y ) , 22 y sec2 y dy = tan y = 3x2 +C , 2 y = tan1 y = 8x3 ey , dy = 3xdx cos2 y 3xdx 3x2 +C 2 y. y (1) = 0 ey dy = 8x3 dx ey = 2x4 + C y = ln 2x4 + C . 1 = 2 + C C = 1 y = ln...

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y = 3x cos2 (y ) , 22 y sec2 y dy = tan y = 3x2 +C , 2 y = tan1 y = 8x3 ey , dy = 3xdx cos2 y 3xdx 3x2 +C 2 y. y (1) = 0 ey dy = 8x3 dx ey = 2x4 + C y = ln 2x4 + C . 1 = 2 + C C = 1 y = ln 2x4 1 . y = y 2 ex x y (2) = 1 2 1 = y dy = y2 t2 e dt 2 2 ex dx x x2 2 et dt + C e dx = 2 y (2) = 1 1 = 0 + C C = 1. 1 =1 y x 2 et dt, 2 y= 1 1 . x t2 e dt 2 y= x2 + 4y 2 x + dy y (1 + y ) = dx x y (0) = 0 y (0) = 0 y (0) = 1 y (0) = 1 C C = 0 x=0 y (1) = 0 y (1) = 0 y=0 C y=0 x dy y = ry ln dt K = ry ln dy = y ln (y/K ) K y y K y +1 . K y = Ke1 y = 0 ln (y/K ) = 1 dy dy y = ry ln = y dt K y ln K y d2 y ry = ry ln 2 dt K t r dt r dt u = ln (y/K ) ; 1 du = dy y du = r dt u ln |u| = rt + C1 ln |ln (y/K )| = rt + C1 |ln (y/K )| = ert+C1 = eC 1 ert = C2 ert ln (y/K ) = C2 ert y = KeC2 e y<K C2 C2 = ln (P0 /K ) y = Keln(P0 /K )e =K K = 10, P0 = 0.5, r = 0.01 C2 = eC1 > 0 rt y (0) = P0 P0 K P0 = eC2 ln (P0 /K ) = K rt ert .
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