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Math Help - Exact Equations

  1. #1
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    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?
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  2. #2
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    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.
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