i f space y equals space left parenthesis x space plus space square root of x squared plus 1 end root right parenthesis to the power of m p r o v e space t h a t space space space space left parenthesis x squared plus 1 right parenthesis fraction numerator d squared y over denominator d x squared end fraction plus space x fraction numerator d y over denominator d x end fraction minus space m squared y equals space 0

Asked by Lochana Jayachandran | 5th Oct, 2014, 07:47: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space f u n c t i o n : y equals open parentheses x plus square root of x squared plus 1 end root close parentheses to the power of m D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma fraction numerator d y over denominator d x end fraction equals m open parentheses x plus square root of x squared plus 1 end root close parentheses to the power of m minus 1 end exponent open parentheses 1 plus fraction numerator x over denominator square root of x squared plus 1 end root end fraction close parentheses equals m open parentheses x plus square root of x squared plus 1 end root close parentheses to the power of m minus 1 end exponent open parentheses fraction numerator x plus square root of x squared plus 1 end root over denominator square root of x squared plus 1 end root end fraction close parentheses equals fraction numerator m open parentheses x plus square root of x squared plus 1 end root close parentheses to the power of m over denominator square root of x squared plus 1 end root end fraction equals fraction numerator m y over denominator square root of x squared plus 1 end root end fraction fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator m squared y square root of x squared plus 1 end root minus m x y over denominator open parentheses x squared plus 1 close parentheses square root of x squared plus 1 end root end fraction open parentheses x squared plus 1 close parentheses fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator m squared y square root of x squared plus 1 end root minus m x y over denominator square root of x squared plus 1 end root end fraction.... left parenthesis 1 right parenthesis x fraction numerator d y over denominator d x end fraction equals fraction numerator m x y over denominator square root of x squared plus 1 end root end fraction.... left parenthesis 2 right parenthesis left parenthesis 1 right parenthesis plus left parenthesis 2 right parenthesis minus m squared y equals m squared y minus fraction numerator m x y over denominator square root of x squared plus 1 end root end fraction plus fraction numerator m x y over denominator square root of x squared plus 1 end root end fraction minus m squared y equals 0 H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 6th Oct, 2014, 10:07: AM