Solve:
$straight y left parenthesis 1 minus straight x squared right parenthesis space dy space plus space straight x space left parenthesis 1 plus straight y squared right parenthesis space dx space equals space 0.$

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Solution

The given differential equation is

straight y left parenthesis 1 minus straight x squared right parenthesis space dy space plus space straight x space left parenthesis 1 plus straight y squared right parenthesis space dx space equals space 0

straight y left parenthesis 1 minus straight x squared right parenthesis space dy space equals space minus straight x left parenthesis 1 plus straight y squared right parenthesis space dx

fraction numerator straight y over denominator 1 plus straight y squared end fraction dy space equals space minus fraction numerator straight x over denominator 1 minus straight x squared end fraction dx

Integrating,

integral fraction numerator straight y over denominator 1 plus straight y squared end fraction dy equals space space minus integral fraction numerator straight x over denominator 1 minus straight x squared end fraction dx

therefore space space space space integral fraction numerator 2 straight y over denominator 1 plus straight y squared end fraction dy space equals space integral space fraction numerator negative 2 straight x over denominator 1 minus straight x squared end fraction dx therefore space space log space left parenthesis 1 plus straight y squared right parenthesis space equals space log space open vertical bar 1 minus straight x squared close vertical bar plus straight c

which is required solution.

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