Question

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

• A. \( \frac{\partial M}{\partial x} = \frac{\partial N}{\partial y} \)
• B. \( \frac{\partial M}{\partial x} = -\frac{\partial N}{\partial y} \)
• C. \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \)
• D. \( \frac{\partial M}{\partial y} = -\frac{\partial N}{\partial x} \)

Hint:

For a differential equation to be exact, the mixed partial derivatives of the functions \( M(x, y) \) and \( N(x, y) \) must be equal: \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). This ensures the existence of a potential function.

Answer 1

The Correct Option is C

The general form of an exact differential equation is: \[ M(x, y) dx + N(x, y) dy = 0 \] For this to be an exact differential equation, the condition is that the mixed partial derivatives of \( M \) and \( N \) must be equal. This means: \[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \] This condition ensures that there exists a potential function \( \Phi(x, y) \) such that: \[ \frac{\partial \Phi}{\partial x} = M(x, y) \quad {and} \quad \frac{\partial \Phi}{\partial y} = N(x, y) \] Let's now analyze each option: - Option (A): \( \frac{\partial M}{\partial x} = \frac{\partial N}{\partial y} \) - This condition is not related to exactness. It is not required for the equation to be exact. Therefore, Option A is incorrect.
- Option (B): \( \frac{\partial M}{\partial x} = -\frac{\partial N}{\partial y} \) - This condition does not satisfy the requirement for exactness. It describes a different kind of relationship between \( M \) and \( N \), not the condition for exactness. Hence, Option B is incorrect.
- Option (C): \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \) - Correct: This is the exact condition for the given differential equation to be exact. When this condition holds, the equation is exact, and a potential function exists. Therefore, Option C is the correct answer.
- Option (D): \( \frac{\partial M}{\partial y} = -\frac{\partial N}{\partial x} \) - This condition is not correct for exactness. It represents a different type of relationship. Thus, Option D is incorrect.
Step 2: Conclusion The correct condition for the given equation to describe an exact differential equation is Option C, where \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).