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Differential Equations

Question
CBSEENMA12033206
Wired Faculty App

Find the particular solution of the differential equation:
                     dy over dx plus straight y space cotx space equals space 2 straight x plus straight x squared space cotx space space space left parenthesis straight x not equal to 0 right parenthesis

Solution
The given differential equation is
                dy over dx plus straight y space cotx space equals space 2 straight x plus straight x squared cotx thin space left parenthesis straight x not equal to 0 right parenthesis
given that y = 0  when x  = straight pi over 2.
                   dy over dx plus straight y space cotx space equals space 2 straight x plus straight x squared space cot space straight x
Comparing it with dy over dx plus Py space equals space straight Q comma we get,
            straight P space equals space cotx comma space space straight Q space equals space 2 straight x plus straight x squared space cotx space space space integral straight P space dx space equals space integral cot space straight x space dx space equals space log space sinx. 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 sinx
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 space 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 straight y space sinx space equals space integral left parenthesis 2 straight x plus straight x squared space cot space straight x right parenthesis space sinx space dx plus straight c or space space space space space space straight y space sinx space equals space integral 2 straight x space sinx space dx space space plus space integral straight x squared space cosx space dx or space space space space space straight y space sinx space equals space left parenthesis sinx right parenthesis. space open parentheses fraction numerator 2 straight x squared over denominator 2 end fraction close parentheses space minus space integral space cos space straight x space open parentheses fraction numerator 2 straight x squared over denominator 2 end fraction close parentheses dx plus space integral space straight x squared space cosx space dx space plus space straight c or space space space space straight y space sinx space equals space straight x squared space sinx space minus space integral straight x squared space cosx space dx space plus space integral straight x squared space cos space straight x space dx space plus space straight c or space space space space straight y space sinx space equals space straight x squared space sin space straight x space plus space 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 space space space space... left parenthesis 1 right parenthesis
Now y = 0  when  straight x space equals space straight pi over 2
therefore space space space space space 0 space equals space straight pi squared over 4 space sin space straight pi over 2 plus straight c space space space space space space rightwards double arrow space space space space 0 space equals space straight pi squared over 4 cross times 1 plus space straight c space space space rightwards double arrow space space space straight c space space equals negative straight pi squared over 4

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