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Differential Equations

Question
CBSEENMA12033166
Wired Faculty App

Solve:
straight x dy over dx plus straight y space equals space straight x space log space straight x

Solution

The given differential equation is straight x dy over dx plus straight y space equals space straight x space log space straight x.
or     dy over dx plus 1 over straight x straight y space equals space log space straight x
Comparing it with dy over dx plus Py space equals space straight Q comma space space we space get comma space space straight P space equals space 1 over straight x comma space space straight Q space equals space log space straight x
                         integral straight P space dx space equals space integral 1 over straight x dx space equals space log space straight x comma space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of logx space equals straight x
therefore space space spacesolution 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 space dx space plus straight c apostrophe space space or space space straight y. space straight x space equals space integral left parenthesis log space straight x right parenthesis. space straight x space dx space plus space straight c space
or        straight y. straight x space equals left parenthesis log space straight x right parenthesis. space straight x squared over 2 minus integral 1 over straight x. straight x squared over 2 dx plus straight c apostrophe space space space or space space straight x space straight y space equals space straight x squared over 2 log space straight x space minus space 1 half integral straight x space dx plus straight c apostrophe
or       straight x space straight y space equals space straight x squared over 2 log space straight x space minus straight x squared over 4 plus straight c apostrophe space space or space space 4 xy space equals space 2 straight x squared logx space minus space straight x squared plus straight c

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