Find a one parameter family of solutions of each of the following differential equation:
y' = e^{x + y} + e^{–x + y}

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Solution

The given differential equation is

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

dy over dx space equals space straight e to the power of straight y left parenthesis straight e to the power of straight x minus straight e to the power of straight x right parenthesis

Separating the variables and integrating, we get,

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

therefore space space space space space space 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 fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction plus straight c space space space space space space space or space space space minus straight e to the power of straight y space equals space straight e to the power of straight x. end exponent space straight e to the power of negative straight x end exponent plus straight c

is the required solutions.

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