यदि `y = sin (ln x)` है, तो निम्नलिखित में से कौन-सा एक सही है ?

Options

A. `(d^(2)y)/(dx^(2)) + y = 0`
B. `(d^(2)y)/(dx^(2)) = 0`
C. `x^(2) (d^(2)y)/(dx^(2)) + x'(dy)/(dx) + y = 0`
D. `x^(2)'(d^(2)y)/(dx^(2)) - x'(dy)/(dx) +y = 0`

Explanation

दिया है,
`y = sin (log_(a) x)`
x के सापेक्ष अवकलन करने पर,
`(dy)/(dx) = (cos (log_(e)x))/(x)`
`x'(dy)/(dx) = cos (log_(e)x)`
पुनः x के सापेक्ष अवकलन करने पर,
`x'(d^(2)y)/(dx^(2)) + (dy)/(dx) = (-sin(log_(e)x))/(x)`
`rArr x^(2)'(d^(2)y)/(dx^(2)) +x'(dy)/(dx)`
`= - sin (log_(e)x)`
`rArr x^(2) (d^(2)y)/(dx^(2)) + x'(dy)/(dx) = - y`
`rArr x^(2)(d^(2)y)/(dx^(2)) + x (dy)/(dx) + y = 0`