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GATE 2025 CH – Question 44

Engineering Mathematics · Differential Equations: First order and higher order equations, Cauchy-Euler equations · 2 marks · Multiple select

Consider the differential equation $\frac{dy}{dx} + \frac{y}{x} = 0$

Choose the CORRECT option(s) for the solution $y$.

  1. $y = x + c$; $c$ is a constant
  2. $y = \frac{c}{x}$; $c$ is a constant
  3. $y = -x + c$; $c$ is a constant
  4. $y = 0$

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Correct answer: (B) $y = \frac{c}{x}$; $c$ is a constant; (D) $y = 0$

Explanation

Separating the variables, $\frac{dy}{y} = -\frac{dx}{x}$, so $\ln y = -\ln x + \text{const}$, which gives $xy = c$, that is $y = \frac{c}{x}$. The choice $c = 0$ gives the trivial solution $y = 0$, which also satisfies the equation. The linear forms in A and C do not satisfy it.