Just started learning about Differential Equations in Khan AP Calc class. I'm wondering if every possible slope field could be said to represent some first order differential equation. Or could you draw a slope field which could not represent one?
If there is some "cheat" such as using piecewise functions to ensure the slopes are always matched, that's worth mentioning, but if that's the case I'm generally interested in what kind of rules would be needed to ensure every drawable slope field had a corresponding differential equation.
I think your question could be rephrased as: What are the minimum requirements for the RHS $f\big(x(t), t \big)$ of the ODE $$\dot x = f\big(x(t), t \big) $$ to posess a solution, right?
For this, there exist two famous theorems, the one due to Peano and the other one to Picard-Lindelöf. As a side note: You also find Integral Curves interesting, if you are asking such questions.