quiz09_s107_solns - y ( t ) = K/ (1 + Ae-kt ) is a solution...

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Math 1B Quiz 9 Solutions Section 107 November 23, 2009 1. (4 points) Find an equation of the curve that passes through the point (0 , - 1) and whose slope at ( x,y ) is x 2 /y . (Give your answer explicity in terms of y .) We are given that dy dx = x 2 y . This is a separable differential equation, so we solve it directly: Z y dy = Z x 2 dx y 2 / 2 = x 3 / 3 + C y 2 = 2 x 3 / 3 + 2 C. By the initial condition, ( - 1) 2 = 0 + 2 C , so y 2 = 2 x 3 / 3 + 1. Since y = - 1 is negative, the final solution is y = - p 2 x 2 / 3 + 1 . 2. (4 points) A colony of penguins is modeled by the differential equation dy dt = ky ± 1 - y K ² where y ( t ) is the biomass of penguins at time t (measured in years), the carrying capacity is estimated to be K = 60 kg, and k = . 05 per year. (a) If y (0) = 15 kg of penguins, estimate how many kg of penguins there will be 1 year later. (Hint: You can use that
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Unformatted text preview: y ( t ) = K/ (1 + Ae-kt ) is a solution to this equation for any A > 0.) We are given an initial condition so we can solve for A : y (0) = 15 = 60 1 + A 1 + A = 60 15 A = 4-1 = 3 Then y (1) = 60 1 + 3 e-. 05 . (b) What is the population of the penguins in the limit (as t goes to innity)? Taking the limit as t goes to innity, we have lim t 60 1 + 3 e-. 05 t = 60 3. (2 points) For each rst order dierential equation below, state whether it is linear, separable, both, or neither. dy dx = yx + 5 y x 2 This one is both. Separable: dy/dx = y [( x +5) /x 2 ]. Linear: dy/dx-[( x +5) /x 2 ] y = 0. dy dx = y 2 + e x This one is neither....
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quiz09_s107_solns - y ( t ) = K/ (1 + Ae-kt ) is a solution...

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