Question

If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?

Answer 1

Here,

\[y = \sin\left( \log x \right)\]

\[\text { Differentiating w . r . t . x, we get }\]

\[\frac{d y}{d x} = \frac{\cos\left( \log x \right)}{x}\]

\[\text { Differentiating again w . r . t . x, we get }\]

\[\frac{d^2 y}{d x^2} = \frac{- \sin\left( \log x \right) - \cos\left( \log x \right)}{x^2}\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = \frac{- \sin\left( \log x \right)}{x^2} - \frac{\cos\left( \log x \right)}{x^2}\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = \frac{- y}{x^2} - \frac{1}{x} \times \frac{dy}{dx}\]

\[ \Rightarrow x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\]