Help with first integral of PDE

[bndnchrs]

Hey guys, I'm having a little difficulty with a pde I'm trying to solve. It boils down to solving for a first integral. I don't want the answer, but I'd be glad to get a little help. We have the system:

$$\frac{dx}{x^2} = \frac{dy}{y^2} = \frac{dz}{xy(z^2 + 1)}$$

We can use the first two and find that

$$\frac{1}{x} - \frac{1}{y} = c$$

I need to use all three to find a second function which is constant here. I've tried using the compendo and dividendo rule and I can't seem to get anywhere... I'm hoping just for a slight hint because I want to solve it myself but I'm really stuck at this point.

Thanks!

 

[ideasrule]

Here are two hints you might find helpful:

(1) You have a relationship between x and y, so you can isolate for x or y, then substitute the result into the original system of equations.

(2) You can manipulate the original system as if the differentials (dx, dy, dz) were parts of a fraction. You can multiply both sides by the same factor, for example, or cancel out common factors, just like with fractions.

 

[bndnchrs]

right... the problem is this solution isn't as easy as all that... there is a more trivial example solved with

$$\frac{dx}{x^2} = \frac{dy}{y^2} = \frac{dz}{z(x+y)}$$

Which can be solved by doing some proper addition and subtraction: so I know the idea. Its a matter of getting a form $$\frac{g*dx + h*dy}{f(x,y)}$$ such that that solution is the differential of a higher form... as the dz part is quite easy, we can integrate it to arctan, we just need a better function of the xs I think...