What are the rules for initial value problems?

[bbb3]

Why is it that you are able to have 1 initial value problem with 2 solutions, but not 2 initial value problems with 1 solution for an ODE?

Is there a theorem that states this?

 

[HallsofIvy]

I have no idea what you are asking. What do you mean by "1 initial value problem with two solutions"? I was under the impression that an initial value problem either had a unique solution or had an infinite number of solutions. (That's basically a statement of the "Fredholm alternative"). On the other hand, as to "not 2 initial value problems with 1 solution for an ODE?", it's easy to make up two distinct intial value problems having the same solution. Is that not what you meant?