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GATE 2024 CE (CE2) – Question 15

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 1 mark · Multiple choice

Consider two Ordinary Differential Equations (ODEs):
P: $\frac{dy}{dx} = \frac{x^4 + 3x^2y^2 + 2y^4}{x^3y}$
Q: $\frac{dy}{dx} = \frac{-y^2}{x^2}$

Which one of the following options is CORRECT?

  1. P is a homogeneous ODE and Q is an exact ODE.
  2. P is a homogeneous ODE and Q is not an exact ODE.
  3. P is a nonhomogeneous ODE and Q is an exact ODE.
  4. P is a nonhomogeneous ODE and Q is not an exact ODE.

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Show answer and explanation

Correct answer: (B) P is a homogeneous ODE and Q is not an exact ODE.

Explanation

In P the numerator and the denominator are both of degree 4, so the right-hand side is a function of $y/x$: P is homogeneous. Q can be written $y^2dx + x^2dy = 0$, with $M = y^2$ and $N = x^2$. Here $\frac{\partial M}{\partial y} = 2y$ and $\frac{\partial N}{\partial x} = 2x$ are not equal, so Q is not exact (it becomes exact only after multiplying by an integrating factor).