# Integrating factor for a non exact differential form

I can't find an integrating factor for the differential form $$-b(x,y)\mathrm{d}x + a(x,y)\mathrm{d}y$$ where $$a(x,y) = 5y^2 - 3x$$ and $$b(x,y) = xy - y^3 + y$$

The problem has origin form the following differential equation \begin{cases} x' = a(x,y) \\ y' = b(x,y) \end{cases} and my teacher told me that an integrating factor for the associated differential form exists.

I have tried to find an integrating factor of the form $\mu(\phi(x,y))$ where $\mu(s)$ is a single variable function.

Requiring $-b(x,y)\mu(\phi(x,y)) \mathrm{d}x + a(x,y)\mu(\phi(x,y)) \mathrm{d}y$ to be closed, I obtained the differential equation

$$\frac{\mathrm{d}\mu(\phi)}{\mathrm{d}\phi} = -\frac{\frac{\partial a}{\partial x} + \frac{\partial b}{\partial y}}{a\frac{\partial \phi}{\partial x} + b\frac{\partial \phi}{\partial y}}\mu(\phi)$$

But I am unable to continue. Any ideas?

## 1 Answer

The integrating factor $\quad \mu=y^2e^x \quad$ can be found thanks to the method below :

The differential relationship leads to a first order linear PDE. We don't need to fully solve it with the method of characteristics. Only a part of the solving is sufficient to find an integrating factor. 