Differential Equations

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Question

Solve: $open parentheses fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction minus fraction numerator straight y over denominator square root of straight x end fraction close parentheses dx over dy equals 1 comma space space space straight x not equal to space 0.$

Solution

The given differential equation is

open parentheses fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction minus fraction numerator straight y over denominator square root of straight x end fraction close parentheses space dx over dy space equals 1 space space space space space space space space space or space space space space space dy over dx space equals fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction minus fraction numerator straight y over denominator square root of straight x end fraction

dy over dx plus fraction numerator 1 over denominator square root of straight x end fraction straight y space equals space fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction

Comparing it with

dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma

straight P space equals fraction numerator 1 over denominator square root of straight x end fraction space space straight Q space equals space fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction

therefore space space space space space integral straight P space dx space equals space integral fraction numerator 1 over denominator square root of straight x end fraction dx space equals space integral straight x to the power of negative 1 half end exponent dx space equals space fraction numerator straight x to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction space equals space 2 square root of straight x

therefore space space space space space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of 2 square root of straight x end exponent

therefore space space space space solution space of space differential space equation space is

straight y. space straight e to the power of integral Pdx end exponent space equals space integral straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight c

straight y space straight e to the power of 2 square root of straight x end exponent space equals integral fraction numerator straight e to the power of negative 2 square root of straight x end exponent over denominator square root of straight x end fraction space straight e to the power of 2 square root of straight x end exponent dx plus straight c

straight y space straight e to the power of 2 square root of straight x end exponent space equals space integral straight x to the power of negative 1 half end exponent dx plus straight c

ye to the power of 2 square root of straight x end exponent space equals space fraction numerator straight x to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction plus straight c space space space space space space space or space space space space straight y space straight e to the power of 2 square root of straight x end exponent space space equals space 2 square root of straight x plus straight c