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Differential Equations

Question

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Solve the differential equation:
$straight x squared dy over dx space equals straight y left parenthesis straight x plus straight y right parenthesis.$

Solution

The given differential equation is

straight x squared dy over dx space equals space straight y left parenthesis straight x plus straight y right parenthesis space space space or space space space space dy over dx space equals fraction numerator straight x space straight y space plus space straight y squared over denominator straight x squared end fraction space space space space space space space space... left parenthesis 1 right parenthesis

Put y = v x so that

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

therefore space space space from space left parenthesis 1 right parenthesis comma space space straight v plus straight x dv over dx space equals space fraction numerator straight x. space straight v space straight x space plus space straight v squared space straight x squared over denominator straight x squared end fraction

or

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

or

straight x dv over dx space equals space straight v squared

Separating the variables, we get,

1 over straight v squared dv space equals space 1 over straight x dx space space space space space or space space space straight v to the power of negative 2 end exponent dv space equals space 1 over straight x dx

Integrating,

integral straight v to the power of negative 2 end exponent space dv space equals space integral 1 over straight x dx

therefore space space space space space space fraction numerator straight v to the power of negative 1 end exponent over denominator negative 1 end fraction space equals space log space open vertical bar straight x close vertical bar space plus straight c space space space space space space or space space space space minus 1 over straight v space equals space log space open vertical bar straight x close vertical bar space plus space straight c therefore space space space space space space fraction numerator negative straight x over denominator straight y end fraction space equals space log space open vertical bar straight x close vertical bar space plus straight c comma space space which space is space required space solution. space

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