Math 2220Problem Set 9Spring 2010Section 3.3:1.For the vector fieldF(x,y) = (−x,y) do the following things:(a) Sketch the vector field.(b) Determine the flow lines by solving the differential equationsdxdt=−xanddydt=y.Your answer should be functionsx(t) andy(t) each of which depends on an arbitraryconstant.(c) Check that the flow lines are hyperbolas by eliminating the variabletfrom yourformulas in part (b).(d) Find the values of the arbitrary constants that give the flow linec(t) such thatc(0) =(2,1).2.Sketch the vector fieldF(x,y) = (x,x2), determine the flow lines, and show the flowlines are parabolas.3.Verify that the parametrized curvec(t) = (sint,cost,2t) is a flow line for the vectorfieldF(x,y,z) = (y,−x,2).4.Consider the vector fieldF(x,y) = (2x,−3).(a) Find a functionf(x,y) whose gradient vector field is equal toF, soF=∇f.(b) Determine the equipotential curves ofF(in other words, the level curves off). Sketcha couple of these curves, and sketch what the vector field
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