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GATE 2017 EC – Question 39

Engineering Mathematics · Differential Equations · 2 marks · Multiple choice

Which one of the following is the general solution of the first order differential equation $\frac{dy}{dx} = (x + y - 1)^2$, where $x, y$ are real?

  1. $y = 1 + x + \tan^{-1}(x + c)$, where $c$ is a constant.
  2. $y = 1 + x + \tan(x + c)$, where $c$ is a constant.
  3. $y = 1 - x + \tan^{-1}(x + c)$, where $c$ is a constant.
  4. $y = 1 - x + \tan(x + c)$, where $c$ is a constant.

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Correct answer: (D) $y = 1 - x + \tan(x + c)$, where $c$ is a constant.

Explanation

Let $u = x + y - 1$. Then $\frac{du}{dx} = 1 + \frac{dy}{dx} = 1 + u^2$. Separating variables gives $\tan^{-1}u = x + c$, so $u = \tan(x + c)$. Substituting back, $y = 1 - x + \tan(x + c)$.