I have the following equations:
$$3x^2 \dfrac{\partial x}{\partial u} + 3y^2 \dfrac{\partial y}{\partial u} = 1$$ $$y\dfrac{\partial x}{\partial u} + x\dfrac{\partial y}{\partial u} = 1$$
where I know that $x\not=y$. I am trying to solve it, but the only way I can is if I divide by $x$ or I divide by $y$, but I have no restrictiont hat these cannot be $0$, except they can't be $0$ simultaneously. Is there a method to solve without dividing by either?
Your equations can be written in the form $$ \begin{pmatrix} 3x^2 & 3y^2 \\ y & x \end{pmatrix}\begin{pmatrix}\dfrac{\partial x}{\partial u}\\ \dfrac{\partial y}{\partial u}\end{pmatrix}=\begin{pmatrix}1\\ 1\end{pmatrix} $$ and so you only need to apply the inverse of the marix on the left-hand side.