Math 310 Q1 Sol - MATH 310 Fall 2011 Differential Equations...

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Unformatted text preview: MATH 310, Fall 2011: Differential Equations Simon Fraser University Quiz 1: Solutions 1. For the differential equation y = 4 + 3 y, sketch the direction field. Hence deduce the behaviour of y ( t ) as t → ∞ ; if this behaviour depends on the initial value y (0) at t = 0, describe the dependency. Solution: Note that y = 0 when 4 + 3 y = 0 , that is, y =- 4 / 3 ; that is, there is an equilib- rium solution y ( t ) ≡ - 4 / 3 . Furthermore, y > when y >- 4 / 3 (positive slopes, so solutions increase away from y =- 4 / 3 ), and y < when y <- 4 / 3 (negative slopes: solutions decrease away from y =- 4 / 3 ). The direction field is thus as follows: Initial conditions at the equilibrium remain there: if y (0) =- 4 / 3 , then y ( t ) =- 4 / 3 for all t . All other solutions, with y 6 =- 4 / 3 , diverge away from the equilibrium (which is unstable): If y (0) >- 4 / 3 , then y ( t ) → + ∞ as t → ∞ ; if y (0) <- 4 / 3 , then y ( t ) → -∞ as t → ∞ . 2. A tank initially contains 1000 litres of water and an unknown amount of salt. Water2....
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Math 310 Q1 Sol - MATH 310 Fall 2011 Differential Equations...

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