यदि ` y= [log (x+ sqrt(x^(2)+1))]^(2)` तो सिद्ध कीजिये की
` " "(x^(2)+1) (d^(3)y)/(dx^(3))+3x (d^(2)y)/(dx^(2))+(dy)/(dx)=0`

Explanation

दिया है- ` " "y= [log (x+ sqrt(x^(2)+1)]^(2)`
दोनों पक्षों का x के सापेक्ष अवकलन करने पर
` (dy)/(dx) =2log (x+sqrt(x^(2)+1))(d)/(dx) log (x+sqrt(x^(2)+1))`
` =2 log (x+ sqrt(x^(2)+1)*(1)/(x+sqrt(x^(2)+1))*(d)/(dx) (x+sqrt(x^(2)+1))`
` =(2log (x+sqrt(x^(2 +1))))/(x+sqrt(x^(2)+1))*(1+(1)/(2sqrt(x^(2)+1))2x)`
` (2log (x+sqrt(x^(2)+_1)))/(x+sqrt(x^(2)+1))*(sqrt(x^(2)+1)+x)/(sqrt(x^(2) +1))`
` =(2log (x+sqrt(x^(2) +1)))/(sqrt(x^(2)+1))`
`rArr" " (sqrt(x^(2)+1))(dy)/(dx) =2log (x+sqrt(x^(2)+1 ))`
` rArr" "(x^(2)+1) ((dy)/(dx) )^(2)=4 [log (x+ sqrt(x^(2)+1 ))]^(2) `
` rArr " "(x^(2)+1) ((dy)/(dx) )^(2)=4y`
अब दोनों पक्षों का x के सापेक्ष अवकलन करने पर
`(x^(2)+1) *2((dy)/(dx) )((d^(2)y)/(dx^(2))) +((dy)/(dx))^(2)*2x =4 (dy)/(dx)`
` rArr " "2(x^(2)+1)(d^(2)y)/(dx^(2))+x (dy)/(dx) =2 `
पुनः दोनों पक्षों का x के सापेक्ष अवकलन करने पर
` [(x^(2)+1)(d^(3)y)/(dx^(3) )+2x (d^(2)y)/(dx^(2)) ]+[(dy)/(dx)+ x(d^(2)y)/(dx^(2))]=0`
` rArr " " (x^(2)+1) (d^(3)y)/(dx^(3))+3x (d^2y)/(dx^(2))+(dy)/(dx) =0`