# solving two systems of equation implicitly

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.

How do you define $xy,xy^2$, when x and y are vectors? –  draks ... Jun 9 '12 at 13:38
had $y$ and $x$ been scalars, i would suggest differentiating the first one wrt $t$, then $$y''=xy^2+x$$ –  Valentin Jun 9 '12 at 13:44
Sorry. my mistake! What I meant was $y = [y1 ;y2 ;y3; ...]$ and $x = [x1; x2;x3; ...]$ and in my system of equations for instance I will have: $\frac{\partial y_{1}}{\partial t}=f(y,x)$ and $\frac{\partial\left(x_{1}y_{1}\right)}{\partial t}=g(y,x)$ , where in g I have terms nonlinear $y^2$ terms. So the squared term would be a dot of for instance $y_1$ and $y_1$. I hope that's more clear. It's basically similar to the Euler equations of flow without the energy equation. –  Hooman Jun 9 '12 at 13:55