Find solution(primitive) of the equation

Question

I want to find its solution of the following equation

$$ydx+xdy+2zdz=0$$

answer:

Keeping $z$ constant;

I obtain that $$ydx+xdy=0$$ or $$\frac{dx}{x}+\frac{dy}{y}=0$$

Then I get $$U(x,y,z)=xy$$

$$\ell= \frac{1}{P}\frac{\partial U}{\partial x}=\frac{1}{y}.y=1$$

Next, $$K=\ell . R -\frac{\partial U}{\partial z}=1. 2z-0=2z\not = 0 $$

So I cannot find its solution. Please help me solving.

Answer 1

Note that $xdy + ydx = d(xy)$ and $2zdz = d(z^2)$. Hence, we get that $$xdy + ydx + 2zdz = d(xy+z^2) = 0$$ This gives us $xy+z^2 = \text{constant}$.