Answer to Question #150541 in Differential Equations for tseb

Question

How to solve
Determine if the following W = (y+z)dx + (x+z)dy + (x+y)dz has a primitive.

Answer 1

Condition, that the given equation (Pfaff equation) has a primitive:

$R(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x})+P(\frac{\partial Q}{\partial z}-\frac{\partial R}{\partial y})+Q(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z})=0$

We have:

$P=y+z, Q=x+z, R=x+y$

Then:

$(x+y)\cdot0+(y+z)\cdot0+(x+z)\cdot0=0$

So, the given equation has a primitive.