Question

Solve $dy over dx plus fraction numerator 1 over denominator straight x space logx end fraction straight y space equals space 1 over straight x.$

Solution

The given differential equation is $dy over dx plus fraction numerator 1 over denominator straight x space logx end fraction. straight y space equals space 1 over straight x$
Comparing it with $dy over dx plus Py space equals space straight Q comma space we space get comma space space straight P space equals space fraction numerator 1 over denominator straight x space logx end fraction comma space straight Q space equals 1 over straight x$
$integral straight P space dx space equals space integral fraction numerator 1 over denominator straight x space logx end fraction dx space equals space integral fraction numerator begin display style 1 over straight x end style dx over denominator log space straight x end fraction space equals space log space left parenthesis log space straight x right parenthesis$
$therefore space space space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space left parenthesis log space straight x right parenthesis end exponent space equals space log space straight x therefore space space space space solution space of space differential space equation space is space space space space space space space space space space space space space straight y space straight e to the power of integral straight P space dx end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dx end exponent space dx space plus space straight c apostrophe space space or space straight y. space log space straight x space equals space integral 1 over straight x. space log space straight x space dx space plus space straight c apostrophe$
or $straight y space log space straight x space equals space integral left parenthesis log space straight x right parenthesis. space 1 over straight x dx plus straight c apostrophe space space or space space straight y space log space straight x space equals space fraction numerator left parenthesis log space straight x right parenthesis squared over denominator 2 end fraction plus straight c apostrophe$
or $2 space straight y space log space straight x space equals space left parenthesis log space straight x right parenthesis squared plus straight c$

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Differential Equations

Solve $dy over dx plus fraction numerator 1 over denominator straight x space logx end fraction straight y space equals space 1 over straight x.$

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