Question

The general solution of the differential equation $dy over dx space equals space straight e to the power of straight x plus straight y end exponent$ is
e^x + e^{– y} = C
e^x+ e^y = C
e^{– x} + e^y = C
e^{– x}+ e^–y = C

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Solution

e^x + e^{– y} = C The given differential equation is

dy over dx space equals space straight e to the power of straight x plus straight y end exponent space space space space space or space space dy over dx space equals space straight e to the power of straight x. space straight e to the power of straight y

Separating the variables, we get,

1 over straight e to the power of straight y dy space equals space straight e to the power of straight x space dx

Integrating,

integral straight e to the power of negative straight y end exponent dy space equals space integral straight e to the power of straight x space dx

therefore space space space space fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction space equals space straight e to the power of straight x plus straight c apostrophe therefore space space space space space space minus straight e to the power of negative straight y end exponent space equals space straight e to the power of straight x plus straight c apostrophe space space or space space straight e to the power of straight x plus straight e to the power of negative straight y end exponent space equals space minus straight c apostrophe therefore space space space space straight e to the power of straight x plus straight e to the power of negative straight y end exponent space equals space straight C comma space which space is space required space solution. space therefore space space left parenthesis straight A right parenthesis space is space correct space answer.

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