The given equation is $$(3x + y - z)p + (x + y - z)q = 2(z -y)$$ where $$p = \frac{\partial z}{\partial x}, q = \frac{\partial z}{\partial y}$$ In my textbook we were prescribed a certain method to solve such equations : applying $ \frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$ where $P = (3x + y - z), Q = (x + y - z), R = 2(z -y)$ but it seems I could not manipulate the above method appropriately.
Generally the solution must come out in the format : $F(u,v) = 0$ where $u$ and $v$ are some functions of $x, y, z$ if we apply this method.My problem is i am not getting the required $u, v$, which matches the book answers.
Note that $x, y$ are independent variables, and $z$ is a dependent variable of $x,y$, and $u,v$ are supposedly expected answers which are themselves functions of $x, y, z$, and so here i need to find the functions $u,v$. Also note that here $p$ is different from $P$ same goes about $q$ and $Q$.
Example of solving thanks to the method of characteristics :
Don't forget that an implicite equation can be written on an infinity of different forms, but all equivalent, that is giving the same general solution, even if the implicite equations look different. So, if your book gives an apparently different equation, the equation found above can be transformed to appear exactly like the equation in the book.