Allied & ICAR Exam Bank • ALLIED_1027 MCQ • 1 Mark

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Exact differential equations

Question Statement:

The differential equation $(2xy + y^3) dx + (x^2 + k x y^2) dy = 0$ is exact if the constant $k$ is:

Options:

  • (A) 1
  • (B) 2
  • (C) 3
  • (D) 4

Answer Key:

C

Step-by-Step Solution & Explanation

For $M dx + N dy = 0$ to be exact, $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. Here $M = 2xy + y^3 \implies \frac{\partial M}{\partial y} = 2x + 3y^2$. $N = x^2 + k x y^2 \implies \frac{\partial N}{\partial x} = 2x + k y^2$. Equating gives $2x + 3y^2 = 2x + k y^2 \implies k = 3$.
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