Question

The necessary and sufficient condition for the differential equation $Mdx + Ndy = 0$ to be exact is

• A. $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial y}$
• B. $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
• C. $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$
• D. $\frac{\partial M}{\partial x} = \frac{\partial N}{\partial x}$

Hint:

To remember this, think "cross-differentiation." Differentiate $M$ (attached to $dx$) with respect to $y$, and differentiate $N$ (attached to $dy$) with respect to $x$. They must match for exactness.

Answer 1

The Correct Option is B

1. Theoretical Foundation: If $M(x, y)dx + N(x, y)dy = 0$ is exact, then there exists a function $f$ such that: $$M = \frac{\partial f}{\partial x} \quad \text{and} \quad N = \frac{\partial f}{\partial y}$$

2. Equality of Mixed Partials: By Clairaut's Theorem (or the symmetry of second derivatives), the mixed partial derivatives must be equal: $$\frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right)$$

3. Resulting Condition: Substituting $M$ and $N$ back into the equality: $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$ This condition ensures that the differential expression $Mdx + Ndy$ is the differential $df$ of some scalar field $f$.