Question

Find the general solution of the following differential equation:

`x (dy)/(dx) = y - xsin(y/x)`

Answer 1

We have the differential equation:

`(dy)/(dx) = y/x - sin(y/x)`

The equation is a homogeneous differential equation.

Putting `y = vx ⇒ (dy)/(dx) = v + x (dv)/(dx)`

The differential equation becomes

`v + x (dv)/(dx) = v - sinv`

⇒ `(dv)/(sinv) = - (dx)/x` ⇒ cosecvdv = `-(dx)/x`

Integrating both sides, we get

`log|"cosec"v - cotv| = - log|x| + logK, K > 0` (Here, log K is an arbitrary constant.)

⇒ `log|("cosec"v - cotv)x| = log K`

⇒ `|("cosec"v - cotv)x| = K`

⇒ `("cosec"v - cotv)x = +- K`

⇒ `("cosec" y/x - cot  y/x)x = C`, which is the required general solution.