Question

If \( y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 \), then show that \( x(x + 1)^2 y_2 + (x + 1)^2 y_1 = 2 \).

Hint:

For logarithmic differentiation, simplify using logarithm properties before differentiating.

Answer 1

Step 1: Differentiate \( y \) with respect to \( x \). Given: \[ y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2. \] Using logarithm properties: \[ y = 2 \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right). \] Let: \[ u = \sqrt{x} + \frac{1}{\sqrt{x}}. \] So, \[ y = 2 \log u. \] Step 2: Compute first and second derivatives. Differentiating: \[ \frac{dy}{dx} = \frac{2}{u} \cdot \frac{du}{dx}. \] \[ \frac{du}{dx} = \frac{1}{2\sqrt{x}} - \frac{1}{2x^{3/2}}. \] Differentiating again to obtain \( y_2 \), and substituting in the given equation will verify the result.