# Vector Field/Differential Equation Correspondence

I have seen some examples (though I am currently looking for a good rigorous explanation and a source would be much appreciated) of taking a second order linear ODE and turning it into a linear system of 2 equations. My question is, can you go the other way? That is, given some continuous vector field, can we find a differential equation corresponding to it (of any order)?

Thanks!

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most differential equations books that ive seen show the solvability of $n$th order linear odes by making a system (and prove existence theorems for systems of first order linear eqns). just create a variable for each derivative, i.e. $x^{(n)}=x_n$ so most of the eqns in the system just look like $x_n=x'_{n-1}$ (not addressing your actual question) – yoyo May 24 '11 at 20:45
I'm not sure I understand your question. Do you want to go from and linear system to a single equation (of any order) ? – Joel Cohen May 24 '11 at 21:30
I mean that for first order and second order ODE's we can get vector fields on $\mathbb{R}^2$ (the first is just a direction field, the second by some massaging, making $y'$ the vertical axis and $y$ the horizontal axis, I believe). However, given a vector field, is it always a solution to some differential equation, of some kind? Can we find that equation? – Jon Beardsley May 25 '11 at 13:14