Can all solutions to differential equations be represented as vector fields?

Question

In all cases, if a differential equation has a solution, can the solution be represented as a vector field? If not, what kind of DEs have solutions that can be represented as vector fields?

Answer 1

Just to clear some things up, here are some terminologies.

A vector field $v:\mathbb{R}^n\rightarrow\mathbb{R}^m$ associates a vector to each point $x\in\mathbb{R}^n$. A scalar field has $m=1$.

I guess, being pedantic, the solution to any DE is a vector field. For instance, the solution to $x'(t) = t$, which is a 1D curve, is a "vector field" with $m=n=1$. It is a little weird though. When $m=1$ and $n=2$ for instance, we have $f(x,y)$ as the solution (a scalar field on $\mathbb{R}^2$); when $m=2$ and $n=1$, we have $g(t)=(x(t),y(t))$, which (to me) is better called a plane curve. But, in a sense, you are associating the vector $g(t)$ to the point $t$, so one could conceivably call it a vector field.

One thing that might be confusing you is the representation of ODEs (not their solutions) as vector fields. Consider $$ \dot{x} = f_1(x,y)\;\;\;\&\;\;\; \dot{y} = f_2(x,y) $$ We can associate a vector $(\dot{x},\dot{y})=(f_1(x,y),f_2(x,y))$ to every point $(x,y)$ on the plane. This is obviously a vector field. Solving the differential equation is essentially starting from an initial condition and generating an integral curve in the plane, i.e. a solution to the DE is a curve whose tangents are given by the vector field $(\dot{x},\dot{y})$.