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Differential Equations

Question

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Solve the following differential equation:
$dy over dx space plus 2 space straight y space tan space straight x space equals space sin space straight x$

Solution

The given differential equation is

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

Comparing it with

dy over dx plus Py space equals space straight Q comma

we get,

straight P space equals space 2 space tanx space space space straight Q space equals space sin space straight x

therefore space space space space integral straight P space dx space equals space 2 space integral space tan space straight x space dx space equals space minus 2 space log space cosx space space equals space log space left parenthesis cosx right parenthesis squared

therefore space space space space space 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 cosx space 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 space space space space space space solution space of space differential space equation space is

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

open square brackets therefore space 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

or

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

or

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

or

straight y equals cosx plus straight c space cos squared straight x

which is required solution.

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