Question

If $y = a \cos(\log x) + b \sin(\log x)$, then $x^2y'' + xy'1$ is:

• A. $\cot(\log x)$
• B. $y$
• C. $-y$
• D. $\tan(\log x)$

Hint:

When differentiating trigonometric functions involving logarithms, remember to apply the chain rule for the logarithmic argument.

Answer 1

The Correct Option is B

The function is $y = a \cos(\log x) + b \sin(\log x)$. The first derivative is $y' = -a \frac{1}{x} \sin(\log x) + b \frac{1}{x} \cos(\log x)$. The second derivative is $y" = -a \frac{d}{dx}\left(\frac{1}{x} \sin(\log x)\right) + b \frac{d}{dx}\left(\frac{1}{x} \cos(\log x)\right)$. Simplifying these derivatives results in the equation $x^2y" + xy'1 = y$.