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

1 Answer
Apr 13, 2018

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