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Differential Equations

Question

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Solve: $open parentheses straight x plus straight y close parentheses space dy over dx space equals space 1.$

Solution

The given differential equation is

open parentheses straight x plus straight y close parentheses dy over dx equals 1 space space space space space space space space space space or space space space straight x plus straight y space equals dy over dx space space space or space space space dx over dy minus straight x space equals space straight y

Comparing it with

dx over dy plus Py space equals straight Q comma space space we space get comma space straight P space equals space minus 1 comma space space straight Q space equals space straight y

integral straight P space dy space equals space minus integral 1 space dy space equals space minus straight y straight e to the power of integral straight P space dy end exponent space equals space straight e to the power of integral space 1 space dy end exponent space equals space straight e to the power of negative straight y end exponent

therefore

solution of given equation is

straight x. space straight e to the power of integral straight P space dy end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dy end exponent space dy space plus straight c

or

straight x. space straight e to the power of negative straight y end exponent space equals space integral straight y. space straight e to the power of negative straight y end exponent dy plus straight c

or space space space space straight x space straight e to the power of negative straight y end exponent space equals space straight y fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction space minus space integral 1. space fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction dy plus straight c or space space space straight x space straight e to the power of negative straight y end exponent space equals space minus ye to the power of negative straight y end exponent plus integral straight e to the power of negative straight y end exponent space dy space plus space straight c or space space space xe to the power of negative straight y end exponent space equals space minus ye to the power of negative straight y end exponent plus fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction plus straight c or space space space xe to the power of negative straight y end exponent space equals space minus ye to the power of negative straight y end exponent plus fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction plus straight c or space space space space xe to the power of negative straight y end exponent space equals space minus ye to the power of negative straight y end exponent minus straight e to the power of negative straight y end exponent plus straight c or space space space space space space space space space straight x space equals space minus straight y space minus space 1 space plus space straight c space straight e to the power of straight y or space space space space straight x plus straight y plus 1 space equals space ce to the power of straight y

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