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Differential Equations

Question

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Solve:
$straight y apostrophe space equals space straight y over straight x plus sin straight y over straight x$

Solution

The given differential equation is

dy over dx space equals space straight y over straight x plus sin space straight y over straight x

Put y = v x so that

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

therefore space space space space from space left parenthesis 1 right parenthesis comma space space straight v plus straight x dv over dx equals straight v plus sin space straight v space space space or space space space straight x dv over dx space equals space sin space straight v

or

fraction numerator 1 over denominator sin space straight v end fraction dv space equals space dx over straight x

or

integral cosec space straight v space dv space equals space integral dx over straight x

or

log space open vertical bar tan space straight v over 2 close vertical bar space equals space log space open vertical bar straight x close vertical bar space plus space log space open vertical bar straight c close vertical bar space space or space space space log space open vertical bar tan space straight v over 2 close vertical bar space equals space log space open vertical bar cx close vertical bar space space or space space tan space straight v over 2 space equals space cx

or

tan space fraction numerator straight y over denominator 2 straight x end fraction space equals space cx space space is space the space required space solution.

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