Differential Equations

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Question

Find one-parameter families of solution curves of the following differential equation:
x y' log x + y = log x

Solution

The given differential equation is

left parenthesis straight x space logx right parenthesis dy over dx plus straight y space equals space log space straight x space space or space space dy over dx plus fraction numerator 1 over denominator straight x space log space straight x end fraction straight y space equals space 1 over straight x

Comparing it with

dy over dx plus straight P space straight y space equals space straight Q comma space we space get comma space straight P space equals space fraction numerator 1 over denominator straight x space logx end fraction comma space space straight Q space equals space 1 over straight x

integral straight P space dx space equals space integral fraction numerator 1 over denominator straight x space log space straight x end fraction dx space equals space integral fraction numerator begin display style 1 over straight x end style dx over denominator log space straight x end fraction space equals space log space left parenthesis log space straight x right parenthesis

therefore space 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 log space straight x right parenthesis end exponent space equals space log space straight x

therefore

solution of differential equation is

straight y space straight e to the power of integral Pdx end exponent space equals space integral straight Q. space straight e to the power of integral straight P end exponent dx space plus space straight c apostrophe space space space or space space straight y. space log space straight x space equals space integral 1 over straight x. space log space straight x space dx plus space straight c apostrophe

straight y space log space straight x space equals integral left parenthesis log space straight x right parenthesis. space 1 over straight x dx plus straight c apostrophe space space or space space straight y space log space straight x space equals space fraction numerator left parenthesis log space straight x right parenthesis squared over denominator 2 end fraction plus straight c apostrophe

2 straight y space log space straight x space equals space left parenthesis log space straight x right parenthesis squared plus straight c