Differential Equations

Solve the following differential equation: . - Wired Faculty

Question

Solve the following differential equation:
$open parentheses 1 plus straight e to the power of straight x over straight y end exponent close parentheses space dx space plus space straight e to the power of straight x over straight y end exponent space open parentheses 1 minus straight x over straight y close parentheses dy space equals space 0$ .

Solution

The given differential equation is

open parentheses 1 plus straight e to the power of straight x over straight y end exponent close parentheses space dx space plus space straight e to the power of straight x over straight y end exponent open parentheses 1 minus straight x over straight y close parentheses dy space equals space 0 space space space space space or space space space dy over dx equals fraction numerator 1 plus straight e to the power of begin display style straight x over straight y end style end exponent over denominator straight e to the power of begin display style straight x over straight y end style end exponent open parentheses begin display style straight x over straight y minus 1 end style close parentheses end fraction space space space space space space space... left parenthesis 1 right parenthesis

Put

straight x space equals space straight v space straight y space space space space or space space dx over dy equals straight v plus straight y dv over dy

therefore space space from space left parenthesis 1 right parenthesis comma space space fraction numerator 1 over denominator straight v plus straight y begin display style dv over dy end style end fraction equals space fraction numerator 1 plus straight e to the power of straight v over denominator straight e to the power of straight v left parenthesis straight v minus 1 right parenthesis end fraction space space space or space space space space straight v plus straight y dv over dy space equals fraction numerator straight e to the power of straight v left parenthesis straight v minus 1 right parenthesis over denominator 1 plus straight e to the power of straight v end fraction

straight y dv over dy space equals fraction numerator straight e to the power of straight v left parenthesis straight v minus 1 right parenthesis over denominator 1 plus straight e to the power of straight v end fraction minus straight v space space equals space fraction numerator ve to the power of straight v minus straight e to the power of straight v minus straight v minus straight v space straight e to the power of straight v over denominator 1 plus straight e to the power of straight v end fraction

straight y dv over dy equals space fraction numerator negative left parenthesis straight e to the power of straight v plus straight v right parenthesis over denominator 1 plus straight e to the power of straight v end fraction space space space or space space space fraction numerator 1 plus straight e to the power of straight v over denominator straight v plus straight e to the power of straight v end fraction dv space equals space minus dy over straight y

integral fraction numerator 1 plus straight e to the power of straight v over denominator straight v plus straight e to the power of straight v end fraction dv space equals space minus integral 1 over straight y dy space space space space space or space space space space log space open vertical bar straight v plus straight e to the power of straight v close vertical bar space equals space minus log space open vertical bar straight y close vertical bar space plus space log space open vertical bar straight c close vertical bar

therefore space space log space open vertical bar straight v plus straight e to the power of straight v close vertical bar space equals space log space open vertical bar straight c over straight y close vertical bar space space space or space space left parenthesis straight v plus straight e to the power of straight v right parenthesis space straight y space equals space straight c space space space space or space space space space open parentheses straight x over straight y plus straight e to the power of straight x over straight y end exponent close parentheses straight y space equals space straight c

straight x plus straight y space straight e to the power of straight x over straight y end exponent equals space straight c space is space the space required space solution. space