Find the general solution of the differential equation:
$dy over dx plus square root of fraction numerator 1 minus straight y squared over denominator 1 minus straight x squared end fraction end root space equals space 0.$

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Solution

The given differential equation is

dy over dx plus square root of fraction numerator 1 minus straight y squared over denominator 1 minus straight x squared end fraction end root space equals space 0

dy over dx space equals space minus fraction numerator square root of 1 minus straight y squared end root over denominator square root of 1 minus straight x squared end root end fraction comma space space or space space space fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction dy space equals space minus fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction dx

Integrating both sides, w.r.t. x,

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

therefore space sin to the power of negative 1 end exponent straight y space equals space minus sin to the power of negative 1 end exponent space straight x space plus space sin to the power of negative 1 end exponent straight c comma space space where space sin to the power of negative 1 end exponent straight c space

is an arbitrary constant.

therefore space space space space sin to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent straight y space equals space sin to the power of negative 1 end exponent straight c

rightwards double arrow space space space space sin to the power of negative 1 end exponent open square brackets straight x square root of 1 minus straight y squared end root plus straight y square root of 1 minus straight x squared end root close square brackets space equals space sin to the power of negative 1 end exponent straight c

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

where c is an arbitrary constant.

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