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GATE 2021 EC – Question 12

Engineering Mathematics · Differential Equations · 1 mark · Multiple choice

Consider the differential equation given below.

$$\frac{dy}{dx}+\frac{x}{1-x^2}\,y=x\sqrt y$$

The integrating factor of the differential equation is

  1. $(1-x^2)^{-3/4}$
  2. $(1-x^2)^{-1/4}$
  3. $(1-x^2)^{-3/2}$
  4. $(1-x^2)^{-1/2}$

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Correct answer: (B) $(1-x^2)^{-1/4}$

Explanation

This is a Bernoulli equation with $n=\frac12$. Substituting $v=y^{1/2}$ gives the linear equation $\frac{dv}{dx}+\frac12\frac{x}{1-x^2}v=\frac x2$. Its integrating factor is $\exp\left(\frac12\int\frac{x}{1-x^2}dx\right)=\exp\left(-\frac14\ln(1-x^2)\right)=(1-x^2)^{-1/4}$.