Given initial and final conditions, find a system of ODEs. Let $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ be two different points in the plane.
Let also
\begin{equation}
\dot{v} = F(v)
\end{equation}
be a system of differential equations in $\mathbb{R}^2$ such that if $v(0)=P_1$ then $v(1)=(1,0)$ and if $v(0)=P_2$ then $v(1)=(-1,0)$. In other words, I'm searching for a system of differential equations that sends the first point to the point $(1,0)$ in a unit time and the second point to the point $(-1,0)$ in a unit time. For simplicity, let's call the points $(1,0)$ and $(-1,0)$ "base points".
I tried searching for a linear system:
\begin{equation}
\dot{v}=Av
\end{equation}
so that the general solution would be
\begin{equation}
v(t) = e^{At}v(0)
\end{equation}
Setting $M:=e^{A\cdot 1}$ and obtaining $M$ from the systems
\begin{equation}
MP_1 = (1,0)\qquad MP_2 = (-1,0)
\end{equation}
one gets (in general) a singular matrix, which makes sense since the base points (vectors) are not independent but in general the points (vectors) $P_1$ and $P_2$ can be independent.
My question is: what could be the "simplest" nonlinear system that gets the job done? Could an affine system of the type $\dot{v}=Av+b$ be enough? I guess so since one can indeed map $P_1$ and $P_2$ to the base points by a rotation, a rescaling and a traslation. But how to achieve this by a system of differential equations?
 A: One "simple" solution uses only straight-line trajectories.
Consider  rectilinear motion in the direction of the unit vector that points from the prescribed given initial point $P_1$ to its prescribed destination point $(1,0)$, and consider likewise a rectilinear path for the second solution. The two lines are generically not parallel, hence intersect in some point $O$ that can be taken as the origin of a polar coordinate grid. Consider therefore the vector field that consists of all unit vector  that radiate toward that central source $O$. Explicitly, $\dot  X = F( X)=- X /|  X |$.
The flow lines of this field are a family of straight lines  that contain the two special lines you want. In order to arrange for both particle trajectories to reach their destinations simultaneously in unit time, some more adjusting is generally needed.  You  need to alter the speeds along these two lines, which can be accomplished by rescaling $\vec F$ by an appropriate scalar factor that  depends only on the polar angle variable $\theta$.
That can be done by interpolating a periodic function $p(\theta)$ that assigns the desired speeds to the two special rays.
The final solution is $\dot X= p(\theta) {\vec X} /| {\vec X} |$.
Note that in some cases the factor $p(\theta)$ may have to be chosen to be negative at some of the rays. (Speed must be interpreted as signed speed along each ray)
