1. For each equation in the following list, determine the set of initial conditions for which the Existence and Uniqueness Theorem guarantees a unique solution. (a) x? = kx (b) x? = sin x (c) x? = t / x (d) x? = cos t + x^3 ? x 2. Let x? = x^(1/3) (a) Show that the hypotheses of the Existence and uniqueness theorem do not hold in any box around the initial point (0, 0). (b) Using an argument similar to that for Torricelli�s law, construct a set of distinct solution curves that all satisfy the initial condition x(0) = 0. (c) Can you prove that the IVP x(0) = 1 has a unique solution? (d) Let x(t0) = x0 ?= 0. Show that the hypotheses of the Existence and uniqueness theorem are satsified if S is a sufficiently small box around (t0,x0). Conclude that the IVP has a unique solution as long as the solu- tion curve lies in S. Find the explicit form of the solution. 3. Let x? = x^2. Show that the hypotheses of the Existence and uniqueness theorem are satisfied for the initial condition x(0) = 1. Calculate the solution and determine the time of the singularity for the solution. Does this contradict the theorem? 4. Let x? =?sqrt(x) with x(0)=2. (a) Show that there exists a unique solution for this initial value problem. (b) Find an explicit solution for the IVP. ??(c) Notice that x1(t) = (sqrt(2) - t/2)^2 0<= t <= sqrt(8) x1(t) = 0 t>sqrt(8) and x2(t) = ( sqrt(2) -t/2 )^2 are two different functions that both seem to solve the differential equation and that both go through the initial point. Does the existence of these two functions imply that the IVP has more than one solution? Explain.