09SpringHwk01 - Math 3301 Homework Set 1 10 Points Basics...

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Math 3301 Homework Set 1 10 Points Sketch the direction field for each of the following differential equations. Based on your direction field sketch determine the behavior of the solution, Basics ( ) y t , as t → ∞ ( i.e. the long term behavior). If this behavior depends upon the value of ( ) 0 y give this dependence. 1. ( ) ( )( ) 2 4 2 2 4 dy y y y dt = + 2. ( ) ( ) 3 2 2 1 y y y ′ = + e For problems 3 & 4 solve the given IVP. Linear Differential Equations 3. ( ) ( ) ( ) 2 2 2 3 2 2 2 0 4 x x y xy x y + = + + = e 4. ( ) ( ) 4 4 3 7 6 3 2 3 1 0 t t y t t y t y ′− + = = e 5. It is known that the solution to the following differential equation will have a relative maximum at 1 2 t = . Assuming that the solution and its derivative exist and are continuous for all t determine the value of the solution at this point ( i.e. find ( ) 1 2 y ). Note that because you don’t have an initial condition you can’t actually solve this differential equation. It is still possible however to answer this question. 6 4 3 t y y ′+ = − e Hint : Recall from Calc I where relative extrema may occur and don’t forget the differential equation,
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