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NDA 2021 Maths · Mathematics · Differential Equations · medium

The integrating factor of (1+x^2dy/dx+2xy=cos x is

Single correct option

The integrating factor of (1+x2)dydx+2xy=cos⁡x(1+x^2)\dfrac{dy}{dx}+2xy=\cos x is
  1. A. 11+x2\dfrac{1}{1+x^2}
  2. B. 1+x21+x^2
  3. C. ex2e^{x^2}
  4. D. ln⁡(1+x2)\ln(1+x^2)
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Answer: (B) 1+x21+x^2

In standard form dydx+2x1+x2y=cos⁡x1+x2\dfrac{dy}{dx}+\dfrac{2x}{1+x^2}y=\dfrac{\cos x}{1+x^2}, so the integrating factor is e∫2x1+x2dx=eln⁡(1+x2)=1+x2e^{\int\frac{2x}{1+x^2}dx}=e^{\ln(1+x^2)}=1+x^2. Indeed the left side of the original equation is ddx[(1+x2)y]\dfrac{d}{dx}[(1+x^2)y].
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Original question written in the NDA 2021 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake