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

Question

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

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Answer 1 · 2018-04-13T15:34:58

Please see below.

#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#