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Differential Equations

Question

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Solve the differential equation:
$left parenthesis 1 minus straight x squared right parenthesis space dy over dx plus straight x space straight y space equals space straight a space straight x$

Solution

The given differential equation is

left parenthesis 1 minus straight x squared right parenthesis space dy over dx plus straight x space straight y space equals space straight a space straight x space space space or space space space dy over dx plus fraction numerator straight x over denominator 1 minus straight x squared end fraction straight y space equals space straight a fraction numerator straight x over denominator 1 minus straight x squared end fraction

Comparing it with

dy over dx plus straight P space straight y space equals space straight Q comma space space we space get

straight P space equals fraction numerator straight x over denominator 1 minus straight x squared end fraction comma space space space space straight Q space equals space fraction numerator straight a space straight x over denominator 1 minus straight x squared end fraction

integral straight P space dx space equals space integral fraction numerator straight x over denominator 1 minus straight x squared end fraction dx space equals space minus 1 half integral fraction numerator negative 2 straight x over denominator 1 minus straight x squared end fraction dx space equals space minus 1 half log space left parenthesis 1 minus straight x squared right parenthesis

therefore space space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of negative 1 half log left parenthesis 1 minus straight x squared right parenthesis end exponent space 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 x squared right parenthesis to the power of negative 1 half end exponent space equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction

Solution of differential equation is

straight y. space fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space equals integral fraction numerator ax over denominator 1 minus straight x squared end fraction. fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction dx plus straight c

or

fraction numerator straight y over denominator square root of 1 minus straight x squared end root end fraction space equals fraction numerator straight a over denominator negative 2 end fraction integral left parenthesis 1 minus straight x squared right parenthesis to the power of negative 3 over 2 end exponent left parenthesis negative 2 straight x right parenthesis space dx plus straight c

or

fraction numerator straight y over denominator square root of 1 minus straight x squared end root end fraction space equals negative straight a over 2 fraction numerator left parenthesis 1 minus straight x squared right parenthesis to the power of negative begin display style 1 half end style end exponent over denominator negative begin display style 1 half end style end fraction plus straight c space space space space or space space space fraction numerator straight y over denominator square root of 1 minus straight x squared end root end fraction space equals fraction numerator straight a over denominator square root of 1 minus straight x squared end root end fraction plus straight c

or

straight y equals straight a plus straight c square root of 1 minus straight x squared end root

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