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I have to solve this differential equation $$ydx+xdy+3x^3y^4dy=0.$$ I'm using Simmons book, and it says that we should set $$M=y$$ and $$N=x+3x^3y^4,$$ then consider $$\dfrac{\partial M}{\partial y}=1,$$ and $$\dfrac{\partial N}{\partial x}=1+9x^2y^4.$$

Now to get the integrating factor we should see if $$\dfrac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x} }{N}=\dfrac{-9x^2y^4}{x+3x^3y^4}$$ or $$\dfrac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x} }{M}=\dfrac{-9x^2y^4}{y}$$ is a function of x and y only.

But this is not the case! So I don´t know what to do to proceed. Can you give some advice please? Or even better, recommend me another book with more detailed examples and methods?

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Consider the change of variables $u=xy$ and $v=y^2/2$. Then the ODE can be written as $du+3u^3dv=0$ or equivalently $\frac{du}{u^3}=-3dv$. Integrate and replace $u=u(x,y)$, $v=v(x,y)$ to obtain $\frac{3y^2}{2}-\frac{2}{x^2y^2}=c$ etc.

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