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Differential Equations

Question
CBSEENMA12033097
Wired Faculty App

Solve:
dy over dx space equals space cos space left parenthesis straight x plus straight y right parenthesis space space or space space cos to the power of negative 1 end exponent open parentheses dy over dx close parentheses space equals space straight x plus straight y

Solution
The given differential equation is
           dy over dx space equals cos left parenthesis straight x plus straight y right parenthesis                       ...(1)
Put x + y = t,       therefore space space space 1 plus dy over dx space equals space dt over dx space space or space space dy over dx space equals dt over dx minus 1
therefore space space from space left parenthesis 1 right parenthesis comma space space dt over dx minus 1 space equals space cos space straight t space space or space space dt over dx equals space 1 plus space cos space straight t therefore space space space space fraction numerator 1 over denominator 1 plus cos space straight t end fraction dt space space equals space dx space space space space rightwards double arrow space space space space integral fraction numerator 1 over denominator 1 plus cos space straight t end fraction dt space equals space integral 1. space dx rightwards double arrow space space space space integral fraction numerator 1 over denominator 1 plus cos space straight t end fraction cross times fraction numerator 1 minus cos space straight t over denominator 1 minus cos space straight t end fraction dt space equals space integral 1. space dx rightwards double arrow space space space space integral fraction numerator 1 minus cos space straight t over denominator 1 minus cos squared straight t end fraction dt space equals space integral 1. space dx space space space rightwards double arrow space space space space integral fraction numerator 1 minus cos space straight t over denominator sin squared straight t end fraction dt space equals space integral 1. space dx rightwards double arrow space integral open parentheses fraction numerator 1 over denominator sin squared straight t end fraction minus fraction numerator cos space straight t over denominator sin squared straight t end fraction close parentheses space dt space equals space integral 1. space dx space space space rightwards double arrow space space space space integral left parenthesis cosec squared straight t minus cosect space cot space straight t right parenthesis space dx space equals space integral 1. space dx rightwards double arrow space space space minus cot space straight t space plus space cosec space straight t space equals straight x plus straight c rightwards double arrow space space space space space fraction numerator cos space straight t over denominator sin space straight t end fraction plus fraction numerator 1 over denominator sin space straight t end fraction space equals space straight x plus straight c space space space space rightwards double arrow space space space space space fraction numerator 1 minus cos space straight t over denominator sin space straight t end fraction space equals space straight x plus straight c rightwards double arrow space space space fraction numerator 2 space sin squared begin display style straight t over 2 end style over denominator 2 space sin begin display style straight t over 2 end style space cos begin display style straight t over 2 end style end fraction space equals space straight x plus straight c space space space rightwards double arrow space space space tan straight t over 2 space equals space straight x plus straight c space space space rightwards double arrow space space space tan fraction numerator straight x plus straight y over denominator 2 end fraction space equals straight x plus straight c space space space which space is space required space solution. space

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