​ Consider the DE xdy - ydx = √(x^2 + y^2)dx (i) Express it in the form dy/dx = f(x,y) (ii) Find the general solution.

Question

Consider the DE

xdy - ydx = \(\sqrt{x^2+y^2} dx\)

(i) Express it in the form \(\frac{dy}{dx} = f(x,y)\)

(ii) Find the general solution.

Answer 1

(i) xdy - ydx = \(\sqrt{x^2 + y^2}dx\)

xdy = (y + \(\sqrt{x^2 + y^2}\)) dx

⇒ \(\frac{dy}{dx}\) = \(\frac{y + ​​\sqrt{x^2 +y^2}}{x}\)

(ii) This is Homogeneous DE

Hence put, y = vx and \(\frac{dy}{dx}\) = v + x\(\frac{dy}{dx}\)