Question

Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
$left parenthesis straight e to the power of straight x plus straight e to the power of straight x right parenthesis space dy space minus space left parenthesis straight e to the power of straight x minus straight e to the power of straight x right parenthesis space dx space equals space 0$

Solution

The given differential equation is

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 dy space equals space left parenthesis straight e to the power of straight x minus 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 space space space space space space space dy space equals space open parentheses fraction numerator straight e to the power of straight x minus straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent end fraction close parentheses dx

Integrating,

integral dy space equals space integral fraction numerator straight e to the power of straight x minus straight e to the power of negative straight x end exponent over denominator straight e to the power of straight x plus straight e to the power of negative straight x end exponent end fraction dx

therefore space space space space space space space space space space space space space space space space space straight y space equals space log space 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 plus straight c comma space space space space straight x space element of space straight R space is space the space solution. space

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