Question

Show that the given differential equation is homogeneous and solve them.

`x dy/dx - y +  x sin (y/x) = 0`

Answer 1

`x dy/dx - y + x sin (y/x) = 0`

`dy/dx = y/x - sin  y/x = g (y/x)`      (say)      ....(i)

The right side of the equation is in the form of `g(y/x)` so it is a homogeneous differential equation of zero degree.

∴Putting  y = vx 

`dy/dx = v + x dy/dx` From equation (i)

`=> v + x (dv)/dx = v - sin v`

`=> x (dv)/dx = v  -  sin v - v`

`=> x (dv)/dx = - sin v`

`=> cosec  v  dv = - 1/x dx`

⇒ log |cosec v - cot v| 

= - log |x| + C₁

On integrating,

⇒ log |(cosec v - cot v)| = C₁

⇒ |x (cosec v - cot v)| = e^C₁

⇒ x (cosec v - cot v) = ± e^C₁ = C           (say)

⇒ `x (cosec y/x - cot y/x) = C`

Which is the required general solution.