Question

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

Solution

The given differential equation is

straight e to the power of straight x dx plus straight e to the power of straight x dy space equals space 0 space space space space space space space space space space space space space space space space or space space space space straight e to the power of straight x dy space equals space minus straight e to the power of straight x dx

therefore space space space space space space straight e to the power of straight y dy space equals space minus straight e to the power of straight x dx

Integration,

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

therefore 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 negative fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction plus straight c subscript 1 space or space space straight e to the power of straight y plus straight e to the power of straight x space equals space straight c comma space space straight x space element of space straight R

is the solution.

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