Differential Equations

Find a particular solution of the differential equation: - Wired Faculty

Question

Find a particular solution of the differential equation:
$dy over dx plus straight y space cotx space equals space 4 straight x$
$cosec space straight x space left parenthesis straight x not equal to 0 right parenthesis comma space space given space that space straight y space equals space 0 space space space when space straight x space equals space straight pi over 2.$

Solution

The given differential equation is

dy over dx plus straight y space cotx space equals space 4 space straight x space cosec space straight x

dy over dx plus left parenthesis cotx right parenthesis. space straight y space equals space 4 space straight x space cosec space straight x

Comparing it with

dy over dx plus Py space equals space straight Q comma space we space get comma

straight P space space equals space cot space straight x comma space space space space straight Q space equals space 4 straight x space cosec space straight x

integral straight P space dx space equals space integral cotx space dx space equals space log space sinx space

therefore space space space space space space space space space 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 log space sinx end exponent space equals space sin space straight x

therefore space

solution of differential equation is

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

or space space space space space space space space straight y space sinx space equals space integral 4 straight x space cosecx. space sinx space dx space plus straight c or space space space space space space space space straight y space sinx space space equals space 4 space integral space straight x space dx space plus space straight c or space space space space space space space space straight y space sinx space space equals space 2 straight x squared plus straight c 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 space space space space space.... left parenthesis 1 right parenthesis Now space straight y space equals space 0 space space space when space straight x space equals space straight pi over 2

therefore space space space space space space space space space space space space space space space space space 0 space equals space 2 cross times straight pi squared over 4 plus straight c space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space straight c space equals space minus straight pi squared over 2

therefore

from (1),

straight y space sinx space equals space 2 straight x squared minus straight pi squared over 2

which is required solution.