Question

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

`x  dy - y  dx =  sqrt(x^2 + y^2)   dx`

Answer 1

`x  dy - y  dx =  sqrt(x^2 + y^2)   dx`,

Which can be written as `x  dy/dx = y + sqrt (x^2 + y^2)`

or `dy/dx = y/x + sqrt (1 + (y/x)^2)`        ....(1)

Since R.H.S. is of the form `g(y/x)`, and so it is a homogeneous function of degree zero.

Therefore equation (1) is a homogeneous differential equation.

To solve this, put y = vx

⇒ `dy/dx = v + x (dv)/dx`

Substituting the value of y and `dy/dx` in (1), we get

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

⇒ `x (dv)/dx = sqrt(1 + v^2)`

⇒ `dx/x = (dv)/sqrt(1 + v^2)`

⇒ `int dx/x = int (dv)/ sqrt(1 + v^2)`

⇒ `log x + log C_1 = log |v + sqrt (1+ v^2)|`

⇒ `log x + log C_1 = log |y/x + sqrt (1 + y^2/x^2)|`

⇒ `log C_1 x = log |y + sqrt (x^2 + y^2)| - log x`

⇒ `pm C_1 x^2 = y + sqrt (x^2 + y^2)`

⇒ `Cx^2 = y + sqrt (x^2 + y^2)`