What are the rules for initial value problems?

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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?
 
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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?
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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