# 3. Definitions of Stability and Boundedness. (a) Assume x = 0 solves the ODE x'= f(t,x), i.e. assume f(1,0) =0, so that x=0 is an equilibrium solution. Explain what it means for this solution to be stable, giving a very precise definition, and then explain what change is made to this definition to say the zero solution is uniformly stable. (b) Explain what it means in general for all solutions of an ODE to be bounded, and then give the definitions of uniform boundedness and ultimate boundedness.

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