Differential Equations

Question

Solve:
$dy over dx plus straight y space secx space equals space tan space straight x$

Solution

The given differential equation is $dy over dx plus straight y space secx space equals space tan space straight x$
Comparing it with $dy over dx plus Py space equals space straight Q comma space we space get comma space straight P space equals space secx comma space space straight Q space equals tan space straight x$
   $integral straight P space dx space equals space integral secx space dx space equals space log space left parenthesis secx plus tanx right parenthesis$
$therefore space space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals straight e to the power of log space left parenthesis secx plus tan space straight x right parenthesis space end exponent space equals space secx space plus tanx$
$therefore$ solution of given equation is
   $straight y left parenthesis secx plus tanx right parenthesis space equals space integral tanx space left parenthesis secx plus tanx right parenthesis space dx plus straight c space open square brackets because space space straight y space straight e to the power of integral Pdx end exponent space equals space integral Qe to the power of integral straight P space dx end exponent plus straight c close square brackets$
   $space equals space integral secx space tan space straight x space dx space plus space integral left parenthesis sec squared straight x minus 1 right parenthesis space dx plus straight c$
or    $straight y space left parenthesis secx space plus space tanx right parenthesis space equals space secx plus tanx minus straight x plus straight c space space space space space space space space space space space space open parentheses straight x not equal to space an space odd space multiple space of space straight pi over 2 close parentheses$

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Differential Equations

Solve:
$dy over dx plus straight y space secx space equals space tan space straight x$

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