How do I find the integrating factor for a differential equation?

[Elen Sakea]

So I've been Stuck On this Equation Trying to find the integrating factor (im not sure if it has one)
appreciate the help

 

[haushofer]

You should write down what you tried. The idea is that your equation can be written as

$$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy = 0$$

for some f(x,y). So apparently your function f(x,y) is constant. Now,

$$\frac{\partial f}{\partial y} = 6xy + 3y^2 x + x^3 , \ \ \ \ \ \frac{\partial f}{\partial x} = 3x^2 + 3y^2$$

You should be able to solve for f(x,y), but without any further information (what did you try, is it an exercise from a book, are there any boundary conditions) we can't help you any further apart from writing out the solution in full. But that's not the idea of this forum ;)

 

[haushofer]

E.g., maybe these notes,

www.math.colostate.edu/~gerhard/M345/CHP/ch2_6.pdf

https://people.clas.ufl.edu/kees/files/ExactDifferentialEquations.pdf

help you further. They contain explicit examples like yours and show you how to find your function f(x,y).