English

If Y = (Sec-1 X )2 , X > 0, Show that X 2 ( X 2 − 1 ) D 2 Y D X 2 + ( 2 X 3 − X ) D Y D X − 2 = 0 - Mathematics

Advertisements
Advertisements

Question

If y = (sec-1 x )2 , x > 0, show that 

`x^2 (x^2 - 1) (d^2 y)/(dx^2) + (2x^3 - x ) dy/dx -2 = 0`

Sum

Solution

y = ( sec-1 x)

`dy/dx = 2 (sec^(-1) x) 1/ (xsqrt(x^2 - 1))`

`x sqrt(x^2 - 1 ) dy/dx = 2 sec^(-1) x`

Again differentiating both sides

`x sqrt(x^2 -1) (d^2 y )/(dx^2) + (dy)/(dx) [sqrt(x^2 - 1 )  +(x^2)/ sqrt(x^2 - 1) ] = (2 xx 1 ) /(x sqrt (x^2 - 1))`

`x sqrt(x^2 -1) (d^2 y )/(dx^2) + (dy)/(dx) ((x^2 - 1 + x^2)/ sqrt(x^2 - 1) ) =  2/(x sqrt (x^2 - 1))`

`[ x (x^2 -1) (d^2 y )/(dx^2) + (dy)/(dx)(2x^2 - 1)] 1/sqrt(x^2 - 1 ) = 2/( x sqrt(x^2 - 1))`

`x^2(x^2 - 1) (d^2y)/(dx^2) + x(2x^2 - 1 ) (dy)/(dx) = 2 `

`x^2(x^2 - 1) (d^2y)/(dx^2) + x(2x^3 - x ) (dy)/(dx) - 2 = 0`

Hence proved.

shaalaa.com
  Is there an error in this question or solution?
2018-2019 (March) 65/3/3

RELATED QUESTIONS

Differentiate `tan^(-1)(sqrt(1-x^2)/x)` with respect to `cos^(-1)(2xsqrt(1-x^2))` ,when `x!=0`


Find the derivative of the following function f(x) w.r.t. x, at x = 1 : 

`f(x)=cos^-1[sin sqrt((1+x)/2)]+x^x`


if `y = sin^(-1)[(6x-4sqrt(1-4x^2))/5]` Find `dy/dx `.


Find `dx/dy` in the following:

`y = cos^(-1) ((2x)/(1+x^2)), -1 < x < 1`


Find `dy/dx` in the following:

`y = sin^(-1)(2xsqrt(1-x^2)), -1/sqrt2 < x  < 1/sqrt2`


Find `dy/dx` in the following:

`y = sec^(-1) (1/(2x^2 - 1)), 0 < x < 1/sqrt2`


Differentiate w.r.t. x the function:

`(sin x - cos x)^(sin x - cos x), pi/4 < x < (3pi)/4`


If `xsqrt(1+y) + y  sqrt(1+x) = 0`, for, −1 < x <1, prove that `dy/dx = 1/(1+ x)^2`


If `sqrt(1-x^2)  + sqrt(1- y^2)` =  a(x − y), show that dy/dx = `sqrt((1-y^2)/(1-x^2))`


Differentiate `tan^(-1) ((1+cosx)/(sin x))` with respect to x


Solve `cos^(-1)(sin cos^(-1)x) = pi/2`


If y = sin-1 x + cos-1x find  `(dy)/(dx)`.


If `log (x^2 + y^2) = 2 tan^-1 (y/x)`, show that `(dy)/(dx) = (x + y)/(x - y)`


If y = `(sin^-1 x)^2,` prove that `(1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.`


`lim_("h" -> 0) (1/("h"^2 sqrt(8 + "h")) - 1/(2"h"))` is equal to ____________.


`lim_("x" -> -3) sqrt("x"^2 + 7 - 4)/("x" + 3)` is equal to ____________.


`lim_("x"-> 0) ("cosec x - cot x")/"x"`  is equal to ____________.


If `"y = sin"^-1 ((sqrt"x" - 1)/(sqrt"x" + 1)) + "sec"^-1 ((sqrt"x" + 1)/(sqrt"x" - 1)), "x" > 0, "then"  "dy"/"dx"` is ____________.


If y `= "cos"^2 ((3"x")/2) - "sin"^2 ((3"x")/2), "then"  ("d"^2"y")/("dx"^2)` is ____________.


If y = sin–1x, then (1 – x2)y2 is equal to ______.


Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then ______.


Differentiate `sec^-1 (1/sqrt(1 - x^2))` w.r.t. `sin^-1 (2xsqrt(1 - x^2))`.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×