# Math Help - Exact Equations

1. ## Exact Equations

So I'm reading through Ordinary Differential Equations by Tenenbaum and Pollard, page 76, Example 9.5, and I find it bizarre. It says that $\cos y dx - (x \sin y - y^{2})dy = 0$ is exact on $\mathbb{R} \times \mathbb{R}$ which I readily believe. It then says we can integrate each part from $(0,0)$ to $(x,y)$, i.e. $\int_{0}^{x}\cos y dx + \int_{0}^{y}y^{2} dy$. The second integrand baffles me. The book hints that, because we're taking the integral with respect to $y$ at the point $(x_{0}, y_{0}) = (0,0)$ with $x$ fixed, we can therefore eliminate the $x \sin y$. But one: Why? Two: If that's so, can't we do the same with the other one and have it be $\int_{0}^{x}(\cos (0))dx = x$?

2. Sorry, just saw that the proof I skipped over because it seemed to assume too much familiarity on my part, gives an explanation of this.