hw1w - 3 V 2 / 3 . The dierential equation becomes, dV dt =...

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Math 2930 8/25/10 solutions problem to be graded - 8 1.1 Match the direction field to the differential equation. Observe that both direction fields have equilibrium solutions (zero slope) at y = 0 and y = 3, which implies choices (e) y 0 = y ( y - 3), and (h) y 0 = y (3 - y ). Note that in 19, y 0 < 0 for y > 3 and therefore corresponds to choice (h). Similarly in 20, y 0 > 0 for y > 3, choice (e). 22 A spherical rain drop evaporates at a rate proportional to its surface area ( S ). Write an differential equation for the volume ( V ) as a function of time. dV dt = r S where r is a rate constant. We know that V = 4 π 3 R 3 and S = 4 πR 2 , which give S = (36 π ) 1 /
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Unformatted text preview: 3 V 2 / 3 . The dierential equation becomes, dV dt = r S = r 2 V 2 / 3 where the constant (36 ) 1 / 3 has been absorbed into r and written as a new constant r 2 . 1.3 8 Verify both functions are solutions to the dierential equation, y 00 + 2 y-3 y = 0 (1) y 1 ( t ) = e-3 t y 1 =-3 e-3 t , y 1 00 = 9 e-3 t (2) Substituting (2) into (1), 9 e-3 t + 2(-3 e-3 t )-3( e-3 t ) = 0 (3) 0 = 0 (4) y 2 ( t ) = e t y 2 = e t , y 2 00 = e t (5) Substituting (5) into (1), e t + 2 e t-3 e t = 0 (6) 0 = 0 (7) 1...
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