How to split a multi-variable ODE into multiple equations?

Question

While thinking quite randomly, a friend of mine came up with the following ODE $$\dot x+\dot y=ax+by$$

Now we want to come up with an equivalent system of ODEs in explicit form, as those can always be found from what we know, but we haven't figured out how to come up with this, eg in the following format with appropriate mappings from the old to the new variables. \begin{align}\dot w&=f(w,z,t)\\\dot z&=g(w,z,t)\end{align}

So my question now is:
Given the first ODE, can we transform it into an explicit system of ODEs, and if so how?
Please don't just throw the transformed version at me, because I want to understand the general / specific approach needed here.

We have thought about a few possible variations, but found no viable transformation. The only one that would technically work would lead to only two equations and three variables being needed, eg you could replace $\dot x=k,\dot y=ax+by-k$ but that would require an extra variable that we would like to avoid.

Answer 1

Generally speaking (not just for differential equations), one can't expect a single equation $A+B = C$ to be equivalent to a system $A=C_1$, $B=C_2$, since the single equation doesn't contain enough information to allow us to say something about $A$ and $B$ individually.

Answer 2

If the ode is considered as an underdetermined system of equations $$ (1,1)\left(\begin{array}{c}x'\\y'\end{array}\right)=(a,b)\left(\begin{array}{c}x\\y\end{array}\right) $$ then the derivatives vector with the least euclidian norm will verify $$ \left(\begin{array}{c}x'\\y'\end{array}\right) = (1,1)^\dagger(a,b)\left(\begin{array}{c}x\\y\end{array}\right) $$ where $(1,1)^\dagger=\frac{1}{2}\left(\begin{array}{c}1\\1\end{array}\right)$ is the pseudo-inverse of $(1,1)$, so $$ \begin{align*} x'&=\frac{1}{2}(ax+by),\\ y'&=\frac{1}{2}(ax+by).\\ \end{align*} $$ So $x$ will be equal to $y$ if $x(0)=y(0)$.