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. dM/dx = dN/dy
• B. dM/dx = - dN/dy
• C. dM/dy = dN/dx
• D. dM/dy = - dN/dx

Hint:

Always check the exactness condition dM/dy = dN/dx before solving. If it fails, you need an integrating factor to make the equation exact.

Answer 1

The Correct Option is C

Step 1: General condition of exactness
A differential equation of the form M(x,y) dx + N(x,y) dy = 0 is called exact if there exists a function F(x,y) such that dF = M dx + N dy. Step 2: Meaning of M and N
This implies M = dF/dx and N = dF/dy. Step 3: Equality of mixed derivatives
If F is continuous and differentiable, then mixed partial derivatives are equal. This gives the condition: dM/dy = dN/dx. Step 4: Checking the options
Option (A): dM/dx = dN/dy is incorrect. Option (B): dM/dx = - dN/dy is incorrect. Option (C): dM/dy = dN/dx is correct. Option (D): dM/dy = - dN/dx is incorrect. \[ \boxed{\text{Correct condition is dM/dy = dN/dx}} \] %Quicktip