Question

If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.

Answer 1

y = x + tan x

∴ `"dy"/"dx" = "d"/"dx"(x + tanx)`
= 1 + sec²x
and
`(d^2y)/(dx^2) = "d"/"dx"(1 + sec x)^2`

= `"d"/"dx"(1) + "d"/"dx"(sec x)^2`

= 2sec x . sec x tan x
= 2 sec²x tan x

∴ `cos^2x.(d^2y)/(dx^2) - 2y + 2x`

= `cos^2x(2sec^2xtanx) - 2(x + tanx) + 2x`

= `cos^2x xx (2)/(cos^2x) xx tan x - 2x - 2tanx + 2x`

= 2 tan x – 2 tan x

∴ `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.