4
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.