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 ( ) yt , as t →∞ ( i.e. the long term behavior). If this behavior depends upon the value of ( ) 0 y give this dependence. 1. ( ) ( )( ) 2 42 2 4 dy y yy dt = + −− 2. ( )( ) 3 22 1 y = −+ e For problems 3 & 4 solve the given IVP. Linear Differential Equations 3. ( ) ( ) ( ) 2 2 23 2 2 2 04 x x y xy x y + = ++ = e 4. ( ) ( ) 4 4 3 76 3 2 3 10 t ty 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 43 t += e Hint : Recall from Calc I where relative extrema may occur and don’t forget the differential equation,
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