Answer Happy

. Consider the scalar ODE 2' = (x + 1)(x - 1)(x - 2)²(x – 3). [One can solve this by sophomore differential equations methods, but doing so it not necessary, or even particularly useful, in answering this question.] Note that this is an ODE to which our elementary arguments show that there is exactly one maximal solution satisfying any initial condition x(to) = 30. (a) What are the equilibrium solutions? (b) Consider an initial condition 2(0) = 20. For which zo will the graph of the solution have a horizontal asymptote as to? (c) Consider an initial condition 2(0) = 80. For which wo will the graph of the solution have a horizontal asymptote as t-00? (d) Consider an initial condition z(0) = xo. For which xo might the graph of the solution have a vertical asymptote? (e) On the same set of axes, graph the solutions satisfying (i) x(0) = 0. (ii) x (2) = 0. (iii) x(0) = 4. (iv) (2) = 4. Putt on the horizontal axis and x on the vertical.