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GATE 2021 ME (ME2) – Question 38

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

Consider the following differential equation $\frac{dy}{dx}=\frac{y}{1+y}$. The solution of the equation that satisfies the condition $y(1)=1$ is

  1. $2ye^y=e^x+e$
  2. $y^2e^y=e^x$
  3. $ye^y=e^x$
  4. $(1+y)e^y=2e^x$

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Show answer and explanation

Correct answer: (C) $ye^y=e^x$

Explanation

Separate variables near the positive initial value: $(1+1/y)dy=dx$. Integrating gives y+ln y=x+C.

The initial condition gives C=0. Exponentiating yields **yeʸ=eˣ (C)**, which also satisfies the original differential equation.