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Differential Equations

Question

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Solve $straight x space dy space minus space straight y space dx space equals space square root of straight x squared plus straight y squared end root space dx.$

Solution

The given differential equation is

straight x space dy minus space straight y space dx space equals space square root of straight x squared plus straight y squared end root space dx

or

straight x space dy space equals space open parentheses straight y plus square root of straight x squared plus straight y squared end root close parentheses dx

or

dy over dx space equals space fraction numerator straight y plus square root of straight x squared plus straight y squared end root over denominator straight x end fraction

Put y = v x so that

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

therefore space space space space left parenthesis 1 right parenthesis space becomes comma space space space space space straight v plus straight x dv over dx space equals space fraction numerator vx plus square root of straight x squared plus straight v squared straight x squared end root over denominator straight x end fraction

or

straight x dv over dx space equals straight v plus square root of 1 plus straight v squared end root minus straight v comma space space space or space space space space straight x dv over dx space equals space square root of 1 plus straight v squared end root

Separating the variables,

fraction numerator 1 over denominator square root of 1 plus straight v squared end root end fraction dv space equals space 1 over straight x dx

Integrating

integral fraction numerator 1 over denominator square root of 1 plus straight v squared end root end fraction dv space equals space integral 1 over straight x dx space space space space space space space space space space space space space space space space rightwards double arrow space space space space log space open vertical bar straight v plus square root of 1 plus straight v squared end root close vertical bar space equals space log space open vertical bar straight x close vertical bar space plus space log space straight c

or

open vertical bar straight v plus square root of 1 plus straight v squared end root close vertical bar space equals space straight c open vertical bar straight x close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space or space space space space space open vertical bar straight y over straight x plus square root of 1 plus straight y squared over straight x squared end root close vertical bar space equals space straight c open vertical bar straight x close vertical bar

or

open vertical bar straight y plus square root of straight x squared plus straight y squared end root space close vertical bar space equals space straight c open vertical bar straight x close vertical bar squared comma space space or space space space space open vertical bar straight y plus square root of straight x squared plus straight y squared end root close vertical bar space equals space straight c space straight x squared

therefore space space straight y plus square root of straight x squared plus straight y squared end root space equals space plus-or-minus space straight c space straight x squared space space space or space space space straight y plus square root of straight x squared plus straight y squared end root space equals space straight A space straight x squared space space where space straight A space equals space plus-or-minus straight c

which is required solution.

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