Question

If y=root1-x/1+x.then prove that (1-X)^2dy/dx+y=0?

If `y=sqrt((1-x)/(1+x))`, then prove that `(1-x^2)(dy)/(dx)+y=0`

Answer 1

Please see below.

Explanation:

`y=sqrt((1-x)/(1+x))`

`y=sqrt((1-x)/(1+x)xx(1-x)/(1-x))`

`y=sqrt((1-x)^2/(1-x^2)`

`y=(1-x)/sqrt(1-x^2)...to(A)`

`"Using "color(blue)"Quotient Rule"`

#(dy)/(dx)=((sqrt(1-x^2))(0-1)-(1-x)(1/(2sqrt(1-x^2)) (-2x)))/((sqrt(1-x^2))^2)#

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

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

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

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

`(1-x^2)(dy)/(dx)=-(1-x)/sqrt(1-x^2)...toUse(A)`

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

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