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"If "y=sqrt((1-x)/(1+x))," prove that "(...

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

Text Solution

Verified by Experts

We have
`y=sqrt((1-x)/(1+x))`
Differentiating w.r.t.x, we get
`(dy)/(dx)=(1)/(2)((1-x)/(1+x))^((1//2)-1)(d)/(dx)((1-x)/(1+x))`
`=(1)/(2)sqrt((1+x)/(1-x))((1+x)(d)/(dx)(1-x)-(1-x)(d)/(dx)(1+x))/(1+x)^(2)=-sqrt((1+x)/(1-x))(1)/((1+x)^(2))`
`"or " (1-x^(2))(dy)/(dx)=-sqrt((1+x)/(1-x))(1)/((1+x)^(2))(1-x^(2))`
`"or " (1-x^(2))(dy)/(dx)=-sqrt((1-x)/(1+x))`
`"or " (1-x^(2))(dy)/(dx)=-y`
`"or " (1-x^(2))(dy)/(dx)+y=0`
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