I have been trying to solve the following two systems of equations simultanously and I'm very hesitant on how to go about it. Whether I need predictor-corrector methods, if I need to linearize the right hand side, etc.
My equations are:
$\frac{\partial y}{\partial t}=yx$
$\frac{\partial(xy)}{\partial t}=xy^2+x$
where, x and y are vectors. I don't want to simplify using $\frac{\partial(xy)}{\partial t}=x\frac{\partial(y)}{\partial t}+y\frac{\partial(x)}{\partial t}$.
I would really appreciate it if someone could explain or name a method that can be used to solve such systems implicitly.I have looked through books ,etc but I seem to be going around in circles and not getting anywhere.
Thanks in advance.