Differential Equations

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Question

Show that the given differential equation is homogeneous and solve it:
$straight y apostrophe space equals space fraction numerator straight x plus straight y over denominator straight x end fraction$

Solution

The given differential equation is

straight y apostrophe space equals space fraction numerator straight x plus straight y over denominator straight x end fraction space space or space space space dy over dx space equals space fraction numerator straight x plus straight y over denominator straight x end fraction

...(1)
It is a differential equation of the form

dy over dx space equals space straight F left parenthesis straight x comma space straight y right parenthesis

Here

straight F left parenthesis straight x comma space straight y right parenthesis space equals space fraction numerator straight x plus straight y over denominator straight x end fraction

Replacing x by

λx space and space straight y space by space λy comma

we get,

straight F left parenthesis λx comma space λy right parenthesis space equals space fraction numerator λx space plus space λy over denominator λx end fraction space equals space fraction numerator straight lambda left parenthesis straight x plus straight y right parenthesis over denominator λx end fraction space equals space straight lambda degree space left square bracket straight F left parenthesis straight x comma space straight y right parenthesis right square bracket

therefore space space space straight F left parenthesis straight x comma space straight y right parenthesis space is space straight a space

homogeneous function of degree zero.

therefore space space given

differential equation is a homogeneous differential equation.
Put y = v x,

therefore space space space dy over dx space equals space straight v. space 1 space plus space straight x dv over dx

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

rightwards double arrow space space straight v plus straight x dv over dx space equals space 1 plus space straight v space space space space space rightwards double arrow space space space straight x dv over dx space equals space 1 space space space space space space space space space space space rightwards double arrow space space space space dv space equals space space 1 over straight x dx

Integrating,

integral space dv space equals space integral 1 over straight x dx comma space space or space space space straight v space equals space log space open vertical bar straight x close vertical bar space plus space straight c

straight y over straight x space equals space space log space open vertical bar straight x close vertical bar space plus space straight c comma space space which space is space required space solution space. space