Answer please

 

Asked by Varsneya Srinivas | 3rd Jan, 2018, 01:42: PM

Expert Answer:

begin mathsize 16px style DE space of space form
straight M left parenthesis straight x comma straight y right parenthesis plus straight N left parenthesis straight x comma straight y right parenthesis dy over dx equals 0 space has space exact space solution space straight F left parenthesis straight x comma straight y right parenthesis space if
Rule space 1 colon space fraction numerator partial differential straight M left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight y end fraction equals fraction numerator partial differential straight N left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight x end fraction.............. left parenthesis straight a right parenthesis
then space there space exists space straight F left parenthesis straight x comma straight y right parenthesis space such space that space
fraction numerator partial differential straight F left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight x end fraction equals straight M left parenthesis straight x comma straight y right parenthesis............ left parenthesis straight b right parenthesis
fraction numerator partial differential straight F left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight y end fraction equals straight N left parenthesis straight x comma straight y right parenthesis............ left parenthesis straight c right parenthesis

So space from space our space question
straight M left parenthesis straight x comma straight y right parenthesis equals 2 xy minus 9 straight x squared
straight N left parenthesis straight x comma straight y right parenthesis equals 2 straight y plus straight x squared plus 1
checking space rule space 1
fraction numerator partial differential straight M left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight y end fraction equals 2 straight x
fraction numerator partial differential straight N left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight x end fraction equals 2 straight x space
so space straight F left parenthesis straight x comma straight y right parenthesis space exists
now space from space left parenthesis straight c right parenthesis
fraction numerator partial differential straight F left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight y end fraction equals straight N left parenthesis straight x comma straight y right parenthesis
straight F left parenthesis straight x comma straight y right parenthesis equals integral space straight N left parenthesis straight x comma straight y right parenthesis space partial differential straight y....... partial space integration
equals integral open parentheses 2 straight y plus straight x squared plus 1 close parentheses space partial differential straight y.
straight F left parenthesis straight x comma straight y right parenthesis equals straight y plus straight x squared straight y plus straight y squared plus straight c left parenthesis straight x right parenthesis.......... constant space of space partial space int space will space contain space terms space of space straight x space
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space since space straight x space was space considered space constant
to space find space straight c left parenthesis straight x right parenthesis
now space from space left parenthesis straight b right parenthesis
fraction numerator partial differential straight F left parenthesis straight x comma straight y right parenthesis over denominator partial differential straight x end fraction equals straight M left parenthesis straight x comma straight y right parenthesis
fraction numerator partial differential over denominator partial differential straight x end fraction open parentheses straight y plus straight x squared straight y plus straight y squared plus straight c left parenthesis straight x right parenthesis close parentheses equals 2 xy minus 9 straight x squared
2 xy plus straight c apostrophe left parenthesis straight x right parenthesis equals 2 xy minus 9 straight x squared
straight c apostrophe left parenthesis straight x right parenthesis equals negative 9 straight x squared
straight c left parenthesis straight x right parenthesis equals negative 3 straight x cubed plus straight c subscript 1
so space straight F left parenthesis straight x comma straight y right parenthesis equals straight y plus straight x squared straight y plus straight y squared minus 3 straight x cubed plus straight c subscript 1


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Answered by Arun | 4th Jan, 2018, 06:58: PM

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