Can pathlines always be found?

Question

Given a two dimensional vector field $$\vec F(x,y) = P(x,y) \hat i+Q(x,y) \hat j $$ To find the path a particle would follow in this flow, one needs to solve the following differential equations: $$\frac {dx}{dt} = P(x,y)$$ $$\frac {dy}{dt} = Q(x,y)$$ Or at least that's what I have found after looking around the internet. However, I have noticed that these equations are not always possible to solve solely in terms of $t$, for example $$\vec F(x,y) = y \hat i - x \hat j $$ giving $$\frac {dx}{dt} = y$$ $$\frac {dy}{dt} = -x$$ Giving solutions that when written as parameters of a parametric equation are not useful because of the $y$ and $x$ terms. I probably am using the wrong methods or I am not fully understanding the concepts. So my question is: Is it possible to find pathlines in any vector field (analytically or numerically), regardless of $P$ and $Q$?

Answer 1

It's always possible numerically, there are many methods to give approximate answers to systems of DEs. Analytically, it's not always possible.

Laplace transforms are my personal favorite way to deal with systems of DEs. I'm not really one for using matrixes if I don't have to. The complete solution to your example is: $x(t) = x(0)\cos(t) + y(0)\sin(t),$ $y(t) = y(0)\cos(t) - x(0)\sin(t)$