I was wondering what this sort of ODE system is classified as, and how to solve it. My understanding is that it should be non-linear, non-homogeneous, non-autonomous, and coupled. Apologies in advance if this is something really simple which i couldn't understand.
$\dot x = \dot y = -rx(t)y(t) $,
where $r$ is a constant.
Thanks!
Since $t$ appears only in $x(t)$ and $y(t)$, the system is autonomous.
If $\dot{x} = \dot{y}$, $y-x$ is constant (let's say $c$). With $y = x+c$ the equation becomes $$ \dot{x} = - r x (x+c)$$ which has general solution $$ x = \frac{c}{A e^{rct} - 1} $$ for arbitrary constant $A$.