For \(|x| \lt 1\), if \(x = \cos\left(\displaystyle\frac{1}{a}\log y\right)\), then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((1-x^2)\displaystyle\frac{d^2y}{dx^2}-x\displaystyle\frac{dy}{dx}-a^2y=0\)
Since \(x = \cos\left(\frac{1}{a}\log y\right)\):
\(\log y = a\cos^{-1}x\)
Differentiating with respect to x:
\(\displaystyle \frac{1}{y}\frac{dy}{dx} = -\frac{a}{\sqrt{1 - x^2}}\)
\(\displaystyle \sqrt{1 - x^2}\,\frac{dy}{dx} = -ay\)
Squaring removes the root:
\(\displaystyle (1 - x^2)\left(\frac{dy}{dx}\right)^2 = a^2y^2\)
Differentiating again, using the product rule on the left:
\(\displaystyle (1 - x^2)\cdot 2\frac{dy}{dx}\frac{d^2y}{dx^2} - 2x\left(\frac{dy}{dx}\right)^2 = 2a^2y\frac{dy}{dx}\)
Dividing by \(2\frac{dy}{dx}\):
\(\displaystyle (1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} - a^2y = 0\)
NCERT (2026-27 edition):
Continuity and Differentiability
