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GATE 2025 MT – Question 36

Engineering Mathematics · Differential equations · 2 marks · Multiple choice

For two continuous functions $M(x,y)$ and $N(x,y)$, the relation $M\,dx+N\,dy=0$ describes an exact differential equation if

  1. $\frac{\partial M}{\partial x}=\frac{\partial N}{\partial y}$
  2. $\frac{\partial M}{\partial x}=-\frac{\partial N}{\partial y}$
  3. $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$
  4. $\frac{\partial M}{\partial y}=-\frac{\partial N}{\partial x}$

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Correct answer: (C) $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

Explanation

If Mdx+Ndy=dF, then M=Fx and N=Fy. Equality of continuous mixed partial derivatives requires My=Nx, which is **C**. This is also locally sufficient on an appropriate simply connected domain with continuous derivatives.