Problem 1 The solution of the exact differential equation 2xy (x2_|_y2)2dx 71 [1 = i 2+ 2)2]dy 0 is a family of curves that can be interpreted as streamlines of a fluid flow around a circular object whose boundary is described by the equation x2 + yz = 1. A slope field with several solution curves for the differential equation is given. a) Solve this DE and denote the implicit solution curves (streamlines) in the form f(x,y) = C. b) Find the streamline corresponding to the initial value (0, %) A container weighing 640 1b will be used to drop supplies from an aircraft into a remote area. Assume the initial velocity of the container is zero as it is dropped from the aircraft. The container is acted upon by two forces, a force due to gravity from the weight of the container and a force due to air resistance with magnitude k|v| ,with &k found empirically to be 1 Ib/(ft/sec). Use g=32 ft/sec? for the constant acceleration due to gravity. Fair a) Let v(t) represent the velocity of the container as a function of time t. Write down the differential equation for the falling container. Write the DE in the form % = f(t,v). b) Find the limiting velocity of the container. ) Ifthe container is likely to burst upon an impact of more than 80 ft/sec, what is the maximum height to which the container can be dropped without likelihood of bursting on impact when 1t reaches the ground? v gravity