+91-8076753736 [email protected]

-->

Differential Equations

Question

Wired Faculty App

Solve: sin (x+y) $dy over dx space equals 1$ .

Solution

The given differential equation is

sin space left parenthesis straight x plus straight y right parenthesis space dy over dx space equals space 1 space space space space space or space space dy over dx space equals space fraction numerator 1 over denominator sin space left parenthesis straight x plus straight y right parenthesis end fraction

...(1)
Put x+ y = t,

therefore space space space 1 plus dy over dx space equals space dt over dx space space space or space space space dy over dx space equals space dt over dx minus 1

therefore space space space space space from space left parenthesis 1 right parenthesis comma space dt over dx minus 1 space equals space fraction numerator 1 over denominator sin space straight t end fraction space space rightwards double arrow space space space dt over dx space equals space 1 plus fraction numerator 1 over denominator sin space straight t end fraction therefore space space space space dt over dx space equals space fraction numerator 1 plus sint over denominator sint end fraction space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space fraction numerator sint over denominator 1 plus sint end fraction dt space equals space dx therefore space space space space integral fraction numerator sin space straight t over denominator 1 plus sin space straight t end fraction dt space equals space integral 1. space dx space space space space rightwards double arrow space space space space integral fraction numerator left parenthesis 1 plus sin space straight t right parenthesis minus 1 over denominator 1 plus sint end fraction dt space equals space integral 1. space dx therefore space space space space integral space open parentheses 1 minus fraction numerator 1 over denominator 1 plus sin space straight t end fraction close parentheses dt space equals space integral 1. space dx rightwards double arrow space space integral open parentheses 1 minus fraction numerator 1 over denominator 1 plus sin space straight t end fraction cross times fraction numerator 1 minus sin space straight t over denominator 1 minus sin space straight t end fraction close parentheses space dt space equals space integral space 1. space dx rightwards double arrow space space integral open parentheses 1 minus fraction numerator 1 minus sin space straight t over denominator 1 minus sin squared straight t end fraction close parentheses space dt space equals space integral 1 space dx space space space space space space space space space rightwards double arrow space space space space space space space integral open parentheses 1 minus fraction numerator 1 minus sin space straight t over denominator cos squared straight t end fraction close parentheses dt space equals space integral 1. dx rightwards double arrow space space space integral open parentheses 1 minus fraction numerator 1 over denominator cos squared straight t end fraction plus fraction numerator 1 over denominator cos space straight t end fraction. fraction numerator sin space straight t over denominator cos space straight t end fraction close parentheses space dt space equals space integral space 1 space. space dx rightwards double arrow space integral left parenthesis 1 minus sec squared straight t space plus space sect space tant right parenthesis space dt space equals space integral 1. space dx rightwards double arrow space space space straight t minus tan space straight t space plus sec space straight t space equals space straight x space plus straight c therefore space space space straight x plus straight y minus tan space left parenthesis straight x plus straight y right parenthesis plus sec left parenthesis straight x plus straight y right parenthesis space equals space straight x plus straight c therefore space space straight y minus tan left parenthesis straight x plus straight y right parenthesis plus sec left parenthesis straight x plus straight y right parenthesis space equals space straight c space is space the space required space solution.

Some More Questions From Differential Equations Chapter

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation