Find a function F(x,y) whose level curves are solutions to the differential equation $$( x^2 - 4xy )dx + x dy= 0$$

I am confused on how to solve this. The idea is to the use exact form to solve it but they don't come out to be exact. I got that

$$M = x^2 - 4xy$$ $$N = x$$

Taking the partial derivative of both does not come out to be exact. What else can I do to solve this?


Multiply your original equation by the integrating factor $\frac{1}{x}e^{-4x}$: \begin{equation} (x^2-4y)e^{-4x}dx + e^{-4x}dy=0. \end{equation} Then $M_y=-4e^{-4x}=N_x$, so this new (and equivalent) differential equation is exact, so you can solve it.

For information on how to come up with the integrating factor, have a look at something like https://math.okstate.edu/people/binegar/2233-S99/2233-l12.pdf.


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