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Vector Algebra

Question

Wired Faculty App

Solve:
$straight y space logydx space minus space straight x space dy space equals space 0$

Solution

The given differential equation is
y log y dx - x dy = 0 or y log y dx = x dy
or

fraction numerator 1 over denominator straight y space log space straight y end fraction dy space equals space 1 over straight x dx

Integrating,

integral fraction numerator begin display style 1 over straight y end style over denominator log space straight y end fraction dy space equals space integral 1 over straight x dx

therefore space space space space space space space space log space open vertical bar log space straight y close vertical bar space equals space log space open vertical bar straight x close vertical bar space plus space log space straight A therefore space space space space space space space space log space open vertical bar log space straight y close vertical bar space minus space log space open vertical bar straight x close vertical bar space equals space log space straight A therefore space space space space log space open parentheses fraction numerator open vertical bar log space straight y close vertical bar over denominator open vertical bar straight x close vertical bar end fraction close parentheses space equals space log space straight A therefore space space space space space space space space space space space space open vertical bar fraction numerator log space straight y over denominator straight x end fraction close vertical bar space equals space straight A space space or space space space space fraction numerator log space straight y over denominator straight x end fraction space equals space plus-or-minus space straight A therefore space space space space space space space space fraction numerator log space straight y over denominator straight x end fraction space equals space straight c comma space space space where space straight c space equals space plus-or-minus straight A or space space space space space space space log space straight y space equals space straight c space straight x space space space space space space space rightwards double arrow space space space space straight y space equals space straight e to the power of cx space is space required space solution.

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