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Differential Equations

Question
CBSEENMA12033095

Solve the differential equation dy over dx space equals sin left parenthesis straight x plus straight y right parenthesis space space or space sin 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 space sin space 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 space dy over dx space equals space dt over dx minus 1
therefore space space space space space space from space left parenthesis 1 right parenthesis comma space space dt over dx minus 1 space equals space space sin space straight t space space space or space space dt over dx space equals space 1 plus sin space straight t
Separating the variables, we get,
                              fraction numerator 1 over denominator 1 plus sin space straight t end fraction dt space equals space dx
Integrating both sides, we get,
                     integral fraction numerator 1 over denominator 1 plus sin space straight t end fraction dt space equals space integral 1. space dx space space space space or space space space space integral 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 dt space equals space integral 1. space dx
or           integral fraction numerator 1 minus sint over denominator 1 minus sin squared straight t end fraction dt space equals space integral 1. space dx space space space space space space space space space space space or space space space space space fraction numerator 1 minus sin space straight t over denominator cos squared straight t end fraction dt space equals space integral 1. space dx
or          integral open parentheses fraction numerator 1 over denominator cos squared straight t end fraction minus fraction numerator sin space straight t over denominator cos squared straight t end fraction close parentheses dt space equals space integral 1. space dx space space or space space integral open parentheses fraction numerator 1 over denominator cos squared straight t end fraction minus 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 dt space equals space integral 1. space dx
or       open parentheses sec squared straight t minus sect space tant close parentheses space dt space equals space 1. space dx
or        tant space minus sect space equals space straight x plus straight c
or      tan left parenthesis straight x plus straight y right parenthesis space minus space sec left parenthesis straight x plus straight y right parenthesis space equals space straight x plus straight c
which is required solution. 


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