ordinary differential equations
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!
I think a number of people have been in a very similar situation to you, in a course on differential equations or otherwise, and have looked for an alternate source for the material. I myself am very familiar with the problem.
There are a number of excellent textbooks on the subject that sell for less that $15, my personal favorite of which is Tenenbaum/Pollard, but this opinion is obviously not objective and you, along with a number of other answeres seem to find significant faults with their development of the subject.
I will list some other possible inexpensive or free resources that I am at least mildly familiar with:
"Bernoulli equations" site:edu filetype:pdf into the google search bar) you will recieve a plethora of lecture notes and descriptions of whatever topic you are looking for. Read one, read 5, read 1000. By reading the lecture notes of many different lecturers you can grasp a topic from a number of different viewpoints and methods of development simultaneously, and this provides an excellent supplement to your course and or any of the resources I described above. I wish you luck with your course. Have fun learning.
If $x(t)=a$ and $y(t)=b$ for some time $t$, then the "arrow" emanating from the point $(a,b)$ points in the direction $(a+b,-a+b)$. This is the vector field to be drawn. For example, using Mathematica,
StreamPlot[{a + b, -a + b}, {a, -4, 4}, {b, -4, 4}]
gives
Best Answer
I assume you're talking about a vector field on $\mathbb{R}^n$ -- indeed, you likely mean a vector field on $\mathbb{R}^2$.
Assuming that's the case, the answer is that if the vector field $\vec{V}(x,y) = v_1(x,y)\hat{i} + v_2(x,y)\hat{j}$ obeys "The Canonical Flip on the Double Tangent Bundle" -- that is to say, if $v_1(x,y) = y$ -- then $\ddot{x} = v_2(x, \dot{x})$ is the second-order ordinary differential equation corresponding to $\vec{V}$.
(For an explanation of the correspondence between ODEs and vector fields, try Ordinary Differential Equations by Arnold. Also, many introductory texts on differential manifolds discuss how tangent vector field are [first-order] ODEs on manifolds; I prefer Differential Manifolds by Kosinski.
To understand The Canonical Flip on the Double Tangent Bundle, I recommend Transversal Mappings and Flows by Abraham and Robbin and The Geometry of Jet Bundles by Saunders.)