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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.

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    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.2012-06-09

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