A1. Consider the initial value problem comprising the ODE
dy/dx= 1 / y²-1
subject to the initial condition.
y(X) = Y,
where X and Y are known constants.
(i) Without solving the problem, decide if (and under what conditions) this initial value problem is guaranteed to have a unique solution. If it does, is the existence of that solution necessarily guaranteed for all values of x?
(ii) Determine the ODE's isoclines, sketch its direction field in the range x € [-3,3] and y € [-3,3]. then sketch a few representative integral curves. [Hint: You should not have to draw the direction field along more than five equally-spaced isoclines, say.] Discuss briefly how the plot of the solution curves relates to the existence and uniqueness results from part (i).
(iii) Find the general solution of the ODE, then apply the initial condition y(0) = 0. You may leave the solution in implicit form.
