How to write Pfaffian differential equation

Question

I am studying the example.

And I dont understand how to write the pfaffian diff equation in the first line.

What is its formula?

Answer 1

For three variables $(x, y, z)$, an equation of the form $P(x, y, z)dx+ Q(x, y, z)dy+ R(x, y, z)dz=0$ is called a Pfaffin DE. A necessary and sufficient condition that a Pfaffin DE is integrable is $X\cdot Curl X= 0$, where $X= (P, Q, R)$. The first one is easy since it is already differential of some function. For the second one if you let $X=(y^2+z^2, xy, xz)$, then $curl X= (0, z, -y)$ and so you will see that $X\cdot Curl X= 0$. Let $x$ treat as constant. Then we have $xydy+xzdz=0$, or $ydy+zdz=0$ and hence $y^2+z^2= U(x, y, z)=c_1$. So there is $\mu$ such that $\mu Q=\frac{\partial U}{\partial y}=2y\Rightarrow \mu=\frac{2}{x}$. We also have $\mu R=\frac{\partial U}{\partial z}$. Then we compute $K=\mu P-\frac{\partial U}{\partial x}=\frac{2U}{x}$. This implies a linear ODE in $U$: $\frac{dU}{dx}+\frac{2U}{x}=0$. Solving this we get $$ U(x, y, z)= x^2(y^2+z^2)=c.$$