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Differential Equations

Question
CBSEENMA12033184

Find a one parameter family of solutions of each of the following differential equation:
straight y apostrophe cos squared straight x space equals space tanx minus space straight y

Solution
The given differential equation is
                    dy over dx. space cos squared straight x space equals space tanx minus straight y space space space space space space space or space space space dy over dx equals tan space straight x space sec squared straight x space minus space straight y space sec squared straight x
or       dy over dx plus left parenthesis sec squared straight x right parenthesis. space straight y space equals space tan space straight x space sec squared straight x
Comparing it with dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma space straight P space equals space sec squared straight x comma space space straight Q space equals tan space straight x space sec squared straight x
                         integral straight P space dx space equals space integral sec squared straight x space dx space equals space tan space straight x comma space space space space space therefore space space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of tanx
therefore solution of differential equation 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
or                 straight y space straight e to the power of tanx space equals space integral tanx space sec squared straight x space. space straight e to the power of tanx dx plus straight c            ...(1)
Let                 straight I space equals space integral tanx space sec squared straight x. space space straight e to the power of tanx dx
  Put  tanx = t,   sec squared dx space equals space dt
therefore space space space space space space space space space space space space space space straight I space equals space integral straight t. space straight e to the power of straight t space dt space equals space straight t space straight e to the power of straight t space minus space integral 1. space straight e to the power of straight t space dt space equals space straight t space straight e to the power of straight t space minus space straight e to the power of straight t space equals space left parenthesis straight t minus 1 right parenthesis space straight e to the power of straight t space equals space left parenthesis tanx space minus space 1 right parenthesis straight e to the power of tanx
therefore space space space space from space left parenthesis 1 right parenthesis comma space space space straight y space straight e to the power of tanx space equals space left parenthesis tanx minus 1 right parenthesis space straight e to the power of tanx plus straight c
or                         straight y equals tan space straight x minus 1 space plus space straight c space straight e to the power of negative tanx end exponent
        which is required solution. 
              

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