Exact De

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The differential equation M dx + N dy = 0 is exact if

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1. **Definition**: An equation is 'exact' if it is the total differential of some function $u(x, y)$, i.e., $du = Mdx + Ndy$.
2. **Symmetry Condition**: For $du = f_x dx + f_y dy$, Clairaut's theorem states that $f_{xy} = f_{yx}$ for continuous functions.
3. **Translation**: Since $M = \frac{\partial u}{\partial x}$ and $N = \frac{\partial u}{\partial y}$, we must have:
   - $\frac{\partial M}{\partial y} = \frac{\partial^2 u}{\partial x \partial y}$ and $\frac{\partial N}{\partial x} = \frac{\partial^2 u}{\partial y \partial x}$.
4. **Conclusion**: The necessary and sufficient condition is **$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$**.

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