Question

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

(x – y) dy – (x + y) dx = 0

Answer 1

(x - y) dy - (x + y) dx = 0

`=> dy/dx = (x + y)/(x - y)`

`= (1 + (y/x))/(1 - (y/x))`

∵ The powers of the numerator and denominator are the same so this is a homogeneous differential equation.

∴ Putting y = vx

From equation (i),

`dy/dx = v + x  (dv)/dx`   

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

`=> x  (dv)/dx = (1 + v)/(1 - v) - v`

`=> x  dy/dx= (1 + v - v + v^2)/(1 - v)`

`=> x  (dv)/dx = (1 + v^2)/(1 - v)`

`=> ((1 - v)/(1 + v^2)) dv = dx/x`

Integrating on both sides

`=> int ((1 - v)/(1 + v^2)) dv = 1/x dx`

`=> int 1/(v^2 + 1) v - 1/2 int (2v)/(v^2 + 1)  dv = int 1/x  dx`

`=> tan^-1 v = 1/2  log (v^2 + 1) + log x + C`

`=> tan^-1 v = 1/2  log (v^2 + 1) + log x + C`

`=> tan^-1  (y/x) = 1/2  log (y^2/x^2 + 1) + log x + C      because y = vx`

`=> tan^-1 (y/x) = 1/2  log ((y^2 + x^2)/x^2) + log x + C`

`=> tan^-1 (y/x) = 1/2  log (x^2 + y^2) - 1/2  log x^2 + log x + C`

`=> tan^-1 (y/x) = 1/2  log (x^2 + y^2) + C`