Question

The given differential equation
$(x y^2 + n x^2 y) dx + (x^3 + x^2 y) dy = 0$ is exact when $n =$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Hint:

To check if a differential equation $Mdx + Ndy = 0$ is exact, verify $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

Answer 1

The Correct Option is D

Step 1: Identify M(x, y) and N(x, y)
Let \[M = x y^2 + n x^2 y \text{and} N = x^3 + x^2 y\] Step 2: Check condition for exactness
Equation is exact if: \[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \] Compute: \[ \frac{\partial M}{\partial y} = 2xy + n x^2 \] \[ \frac{\partial N}{\partial x} = 3x^2 + 2x y \] Equating: \[ 2xy + n x^2 = 3x^2 + 2x y \Rightarrow n x^2 = 3x^2 \Rightarrow n = 3 \] Answer: 3