I've been reading Hirsch and Smale's Differential equations, dynamical systems and linear algebra.
The author mentions that it is possible to sketch the graph of a solution to a differential equation in two dimensions (say, $x' = f(t,x)$). This makes sense to me since we obtain a function $x(t)$ and we can simply graph this function.
However, the text then says that we can graph a solution to a system of two differential equations in 3 dimensions. I don't quite understand how this is possible. We will get as a solution to our system two functions, say $x(t)$ and $y(t)$. How can we plot these in three dimensions, since they both depend on the independent variable $t$? Does the author mean that we can plot the first solution in the $xt$ plane and the second in the $yt$ plane?
Best Answer
No. It means that you can graph the (parametric) curve $(t,x(t),y(t))$ in $txy$ space. If the equation is autonomous, you can project those curves onto the $XY$ plane (the so called phase plane) because they are invariant under $t$-translations.