1.

यदि `y=sqrt((1-x)/(1+x))`, तब `(1-x^(2))(dy)/(dx)+y,-1ltxlt1`. का मान ज्ञात कीजिए

Answer» दिया गया है-
`y=sqrt((1-x)/(1+x))=((1-x)/(1+x))^(1/2)`
दोनों पक्षों का x के सापेक्ष अवकलन करने पर,
`(dy)/(dx)=d/(dx)((1-x)/(1+x))^(1/2)`
`=1/2((1-x)/(1+x))^((-1)/2)d/(dx)((1-x)/(1+x))`
`=1/2((1-x)/(1+x))^((-1)/2){((1+x)d/(dx)(1-x)-(1-x)d/(dx)(1+x))/((1+x)^(2))}`
`=(sqrt(1+x))/(2sqrt(1-x)){((1+x)(0-1)-(1-x)(0+1))/((1+x)^(2))}`
`=(sqrt(1+x))/(2sqrt(1-x))xx(-2)/((1+x)^(2))`
`=(-1)/(sqrt(1-x)(1+x)^(3//2))` ...(1)
अब, `(1-x^(2))(dy)/(dx)+y`
`=(1-x)^(2)xx(-1)/(sqrt(1-x)(1+x)^(3//2))+sqrt((1-x)/(1+x))`
`=(-1(1-x)(1+x))/(sqrt(1-x)(1+x)^(3//2))+sqrt((1-x)/(1+x))`
`=-sqrt((1-x)/(1+x))+sqrt((1-x)/(1+x))`
=0


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