Question

Consider an autonomous ODE y = f(y) where f(y) = y2 - 1 A. (1 pt) Draw the graph of function f(y) in the y,y plane and specify the points y where f(y) is singular (that is, f(y) takes an infinite value). B. (1 pt) Finf the equilibrium solution and determine its type: stable, un- stable or semi-stable. Indicate on which side it attracts/repels nearby solutions. C. (2 pt) By separating y and t, specify the general solution y=yt,C) t+C /(t+C)2 where C is the constant of integration. A: Y=2 fV4 D. (1 pt) Determine the intervals of existence of a solution y(t,C) on the t-axis for a given C. A: L: - <t<-C-2, L : 2-C<t<. E. (2 pt) Show the graph(s) of solution(s) y(t, C) for C = 0, including the arrows along the curves. Hint: You may first draw a graph where t is a function of y and rotate the picture by a direct angle. F. (1 pt) Show the directional field over the lines y = 0, £0.5, +1.05, +2. G. (2 pt) For the IVP with y(0) = 0.5, identify the solution y(t), indicate the interval I where y(t) exists, determine the range of values for y(t) and the long-time limit y() = lim y(t). A: I: -0.5 <t< , y(t) with t, from y(-0.5) = 1 to y(0) = 0. t-00

Answer 1

Graph 10 5 0 -5 -10 -5 5 -10 0 1 X -(y^2)/(y^2-1) = + 11 TO

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