Question

The integrating factor of the differential equation:
$left parenthesis 1 minus straight y squared right parenthesis dx over dy plus straight y space straight x space equals space straight a space straight y space space left parenthesis negative 1 less than straight y less than 1 right parenthesis$

$fraction numerator 1 over denominator straight y squared minus 1 end fraction$
$fraction numerator 1 over denominator square root of straight y squared minus 1 end root end fraction$
$fraction numerator 1 over denominator 1 minus straight y squared end fraction$
$fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction$

Solution

fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction

The given differential equation is $left parenthesis 1 minus straight y squared right parenthesis space dx over dy plus space straight y space straight x space equals space straight a space straight y$
or $dx over dy plus fraction numerator straight y over denominator 1 minus straight y squared end fraction straight x space equals space fraction numerator straight a space straight y over denominator 1 minus straight y squared end fraction$
Comparing it with $dx over dy plus Px space equals space straight Q comma space space we space get comma space space straight P space equals space fraction numerator straight y over denominator 1 minus straight y squared end fraction comma space space space straight Q space equals space fraction numerator straight a space straight y over denominator 1 minus straight y squared end fraction$
I.F. = $straight e to the power of integral straight P space dy end exponent space equals space straight e to the power of integral fraction numerator straight y over denominator 1 minus straight x squared end fraction dy end exponent space equals space straight e to the power of 1 half integral subscript 1 minus straight y squared end subscript superscript negative 2 straight y end superscript dy end exponent space equals space straight e to the power of negative 1 half log space left parenthesis 1 minus straight x squared right parenthesis end exponent$
$equals space straight e to the power of log space left parenthesis 1 minus straight x squared right parenthesis to the power of negative 1 half end exponent end exponent space equals space left parenthesis 1 minus straight y squared right parenthesis to the power of negative 1 half end exponent space equals space fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction$
$therefore space space space space left parenthesis straight D right parenthesis space is space correct space answer.$

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