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Differential Equations

Question

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Solve:
$straight x space log space straight x space dy over dx plus straight y space equals space 2 over straight x log space straight x$

Solution

The given differential equation is

straight x space log space straight x space dy over dx plus straight y space equals space 2 over straight x log space straight x space space space or space space space dy over dx plus space fraction numerator 1 over denominator straight x space log space straight x end fraction straight y space equals space 2 over straight x squared

Comparing it with

dy over dx plus Py space equals straight Q comma space we space get comma space space straight P space equals fraction numerator 1 over denominator straight x space log space straight x end fraction comma space straight Q space equals 2 over straight x squared

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

therefore

solution of given 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 space plus straight c

or

straight y space log space straight x space equals space integral 2 over straight x squared log space straight x space dx space plus space straight c

or

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

or space space space straight y space log space straight x space equals space 2 open square brackets left parenthesis log space straight x right parenthesis. space fraction numerator straight x to the power of negative 1 end exponent over denominator negative 1 end fraction minus integral 1 over straight x. fraction numerator straight x to the power of negative 1 end exponent over denominator negative 1 end fraction dx close square brackets plus straight c or space space straight y space log space straight x space equals space minus fraction numerator 2 space log space straight x over denominator straight x end fraction plus 2 space integral straight x to the power of negative 2 end exponent space dx plus straight c or space space straight y space logx space space equals space minus fraction numerator 2 space log space straight x over denominator straight x end fraction plus 2 fraction numerator straight x to the power of negative 1 end exponent over denominator negative 1 end fraction plus straight c or space space space straight y space logx space equals negative 2 over straight x left parenthesis 1 plus log space straight x right parenthesis space plus space straight c comma space which space is space required space solution.

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