Differential Equations

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Question

For the given differential equation, find a particular solution satisfying the given condition:
$dy over dx plus 2 straight y space tan space straight x space equals space sin space straight x space colon space straight y space space equals space 0 space space when space straight x space equals space straight pi over 3$

Solution

The given equation is

dy over dx plus 2 straight y space tan space straight x space equals space sin space straight x

Comparing with

dy over dx plus straight P space straight y space equals space straight Q comma space space we space get comma space space straight P space equals space 2 space tanx space comma space space straight Q space equals space sin space straight x

therefore space space space space space integral straight P space dx space equals space integral 2 space tan space straight x space dx space equals space minus 2 space log space cosx space equals space log space open parentheses cosx close parentheses squared therefore space space space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space left parenthesis cos space straight x right parenthesis to the power of negative 2 end exponent end exponent space equals space left parenthesis cos space straight x right parenthesis to the power of negative 2 end exponent space equals space fraction numerator 1 over denominator cos squared straight x end fraction

therefore

solution of given equation is

straight y. space fraction numerator 1 over denominator cos squared straight x end fraction space equals space integral space sinx. space space fraction numerator 1 over denominator cos squared straight x end fraction dx plus straight c

open square brackets because space space straight y space straight e to the power of integral straight P space dx 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 close square brackets

fraction numerator straight y over denominator cos squared straight x end fraction space equals space integral secx space tanx space dx space plus space straight c

or space space space fraction numerator straight y over denominator cos squared straight x end fraction space equals secx space plus space straight c or space space space straight y space equals space cos space straight x space plus space straight c space cos to the power of 2 straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis because space space space space space space straight y space equals space 0 space space space space when space straight x space equals space straight pi over 3 therefore space space space space space space 0 space equals space 1 half plus straight c space open parentheses 1 half close parentheses squared space space space space space space space space space space space therefore space space space straight c space equals space minus 2 therefore space space space from space left parenthesis 1 right parenthesis comma space space straight y space equals space cosx minus 2 space cos squared straight x.