Solve the following differential equation:
$left parenthesis straight y plus straight x space straight y right parenthesis space dx space plus space left parenthesis straight x minus straight x space straight y squared right parenthesis space dy space equals space 0$

Solution

The given differential equation is
$left parenthesis straight y plus straight x space straight y right parenthesis space dx space plus space left parenthesis straight x minus straight x space straight y squared right parenthesis space dy space equals space 0$
$or space straight y left parenthesis 1 plus straight x right parenthesis space dx space plus space straight x left parenthesis 1 minus straight y squared right parenthesis space dy space equals space 0 or space space straight x left parenthesis 1 minus straight y squared right parenthesis space dy space equals space minus straight y left parenthesis 1 plus straight x right parenthesis space dx or space fraction numerator 1 minus straight y squared over denominator straight y end fraction dy space equals space minus fraction numerator 1 plus straight x over denominator straight x end fraction dx$
Integrating, $integral fraction numerator 1 minus straight y squared over denominator straight y end fraction dy space equals space minus fraction numerator 1 plus straight x over denominator straight x end fraction dx$
$therefore space space space integral open parentheses 1 over straight y minus straight y close parentheses space dy space equals space minus integral open parentheses 1 over straight x plus 1 close parentheses space dx therefore space space log space open vertical bar straight y close vertical bar minus straight y squared over 2 space equals space minus log space straight x space minus space straight x space plus straight c comma space which space is space the space required space solution. space$

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Vector Algebra

Solve the following differential equation:
$left parenthesis straight y plus straight x space straight y right parenthesis space dx space plus space left parenthesis straight x minus straight x space straight y squared right parenthesis space dy space equals space 0$

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