Suppose the hypotheses of the existence theorem are satisfied for dy/dt-f(t,y) at the point (t_0,y_0). Then you are guaranteed a solution exists. For how long in time can you be sure it exists? If the hypotheses of the existence theorem fail at some point, can you be sure that a solution doesn't exist there? Can two solutions to a differential equation ever cross? Explain why not or when this might be possible. Give an example of a situation in which you can be sure a solution to a differential equation will last forever.