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Let us consider the vector field $X=x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}$. I'd like to compute the integral curves. Solving the system of ODE

$\begin{cases} x'(t)=x(t) \\ y'(t)=y(t) \end{cases}$

we clearly get the curves $(c_1e^t,c_2e^t)$. Using implicit equations, we get $c_1y-c_2x=0$, i.e. the integral curves are lines through the origin and vector field is the usual radial field (think for instance of electrostatic field generated by a single charge). However, I have a problem in the interpretation of the result: by the Cauchy's existence and uniqueness theorem, I expect that two integral curves cannot intersect; so, how is it possible to obtain straight lines through the origin (and hence intersecting)? Which is the mistake in my argument?

TheWanderer
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1 Answers1

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The origin is an equilibrium, i.e. $x'(0)=y'(0)=0$. Thus, by the uniqueness theorem, no trajectory starting outside of the origin can reach $(0,0)$ in finite time. However, in the limit for $t\rightarrow-\infty$, every trajectory of your system converges towards $(0,0)$.

Since the implicit characterization of trajectories does not contain any time dependence, this information gets lost in the geometric picture of $c_2x-c_1y=0$.

Nuke_Gunray
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