Show that (x2 -y2) die + 2xy dy = 0 is a differential equation of the type dy/dx =g(y/x).

Question: Show that (x2 -y2) die + 2xy dy = 0 is a differential equation of the type dy/dx =g(y/x).

 

The correct answer is –

To show that (x2 – y2) dy + 2xy dx = 0 is a differential equation of the type dy/dx = g(y/x), we can make the substitution y = vx, which gives us dy/dx = v + x (dv/dx). We can also express x in terms of v and y as x = y/v. Substituting these values in the given equation, we get:

(x2 – y2) dy + 2xy dx = 0 (x2 – v2x2) (v dx) + 2y (y/v) dx = 0 (x2 – v2x2) v + 2y (y/v) (dv/dx) = 0 (v2 – 1) x2 dv + 2y2 v dv = 0

Dividing both sides by x2v, we get:

(v2 – 1) dv + 2(v/y) dv = 0

This is a differential equation of the type dy/dx = g(y/x), where g(y/x) = -2(y/x)/ (1 – (y/x)2).