I have been trying to solve the following problem, but I am getting stuck...  we have: $-y^2dx + x^2dy = 0$   given: $u(x,y) = \frac{1}{(x-y)^2}$  both have partial derivative: $\frac{-2xy}{(x-y)^3}$  so we have the exact equation: $\frac{-y^2}{(x-y)^2}dx + \frac{x^2}{(x-y)^2}dy = 0$  now, I try integrate the dx expression: $\int\frac{-y^2}{(x-y)^2}dx$  This gives me: $\frac{y^2}{(x-y)} + g(y)$  Differentiating with respect to y gives: $\frac{-y(y-2x)}{(y-x)^2} +g'(y)$   I have checked my work for the above integral and derivative and also used an online calculator to verify that they are correct. Every other problem of this type that I have tried has had something meaningful come of:  $\frac{-y(y-2x)}{(y-x)^2} +g'(y) = \frac{x^2}{(x-y)^2}$  However, I have not been able to get anywhere using this to try to find a value for $g(y)$
I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is:  $-y^2dx +x^2dy = 0$  He then asks us to verify that:  $ u(x, y) = \frac{1}{(x-y)^2}$  is an integrating factor.   I multiplied through to get:  $\frac{-y^2}{(x-y)^2}dx + \frac{x^2}{(x-y)^2}dy = 0$   However, the partial derivatives of these do not equal each other so I am a bit confused...
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How to show this property of $\mathbb{RP}^{n}$. To prove that: The $n$-dimensional projective space can be presented as:  $\mathbb{RP}^n=\bigcup_{i=0}^{n} A^{i}$.  (The intersection of each two $A^{i}$ is empty)، $A^{i}$ is isomorphic to $\mathbb R^{i}$ ($A^{0}$ includes only one point). The definition of $\mathbb{RP}^n$ is: $\mathbb{RP}^n= (\mathbb R^{n+1} \setminus \{0\})/ \sim$ , $\sim$ is defined as: $x \sim \lambda x$for all $\lambda\in \mathbb R^\times$ and $x\in \mathbb R^{n+1} \setminus \{0\}$. $\mathbb R^\times $ is the productive group of scalars $\ne0$ in the field $\mathbb R$. I'm just a beginner in this field so any help would be appreciated. How to return a part of an array in Ruby? How to add right-click (on desktop) to create text file functionality 16.04.2 Ubuntu GNOME How to close ports / disable services? 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