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GATE 2020 EC – Question 37

Engineering Mathematics · Differential Equations · 2 marks · Multiple choice

Which one of the following options contains two solutions of the differential equation $\dfrac{dy}{dx}=(y-1)x$?

  1. $\ln|y-1|=0.5x^2+C$ and $y=1$
  2. $\ln|y-1|=2x^2+C$ and $y=1$
  3. $\ln|y-1|=0.5x^2+C$ and $y=-1$
  4. $\ln|y-1|=2x^2+C$ and $y=-1$

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Show answer and explanation

Correct answer: (A) $\ln|y-1|=0.5x^2+C$ and $y=1$

Explanation

Separating variables, $\frac{dy}{y-1}=x\,dx$ gives $\ln|y-1|=\frac{x^2}{2}+C$. The constant function $y=1$ also satisfies the equation, since both sides are zero. $y=-1$ does not.