Question

For the differential equation find a particular solution satisfying the given condition:

`dy/dx -  y/x + cosec (y/x) = 0; y = 0` when x = 1

Answer 1

Given differential equation

`dy/dx - y/x + cosec (y/x)` = 0              ....(i)

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

Clearly, this equation is a differential equation.

∴ Putting y = vx

`v + x (dv)/dx` = v - cosec  v in equation (i)

`=> x  (dv)/dx` = - cosec v

sin v dv = `- 1/x` dx

On integrating,

`- int sin  v  dv = int 1/x dx`

cos v = log x + C

So on putting `(y/x)` in place of v,

`cos (y/x) = log |x| + C`     ....(ii)

Given y = 0 if x = 1              ...[from equation (ii)]

cos 0 = log 1 + C

⇒ C = 1

Putting this value of C in equation (ii),

`cos (y/x) = log |x| + log |e|`

`cos (y/x) = log  |ex|`