MATH 024 Midterm: math24_2009s_exam1_soln_0 (1)
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Lecturer: boaz ilan: (30 points) consider the equation dy dt ty. 2+ ty (a) define what an equilibrium solution is in general. (b) find the equilibrium solutions of the equation above. (c) sketch the direction field and determine the stability of the equilibrium solutions. (d) find the general solution. What is its long-time behavior? (e) perform the transformation tv and classify this equation. Solution: (a) an equilibrium solution is a constant solution, that is ty c. Its graph the (t, y) plane is a horizontal line. (b) equilibrium solutions can be found by setting f (t, y ) = 0. = ty + ty2 = 0 ty(y +1) = 0. Y = 0 and y = -1 are equilibrium solutions. Note that t = 0 is not an equilibrium solution or a solution at all. (c) the direction field has been plotted below.
