Question

Find the equation of the curve passing through (3, 4) and satisfying the differential equation y(dy/dx)² + (x – y)(dy/dx) – x = 0.

Answer 1

The given differential equation is

yp² + (x – y)p – x = 0, where dy/dx = p

yp² – yp + xp – x = 0

yp(p – 1) + x(p – 1) = 0

(yp – 1)(p – 1) = 0 

(yp + x) = 0, (p – 1) = 0

xdx + ydy = 0, dy = dx

Integrating, we get 

x² + y² = a², y = x + b

which is passing through (3, 4), so a² = 25 and b = 1 

Hence, the equations of the curves are

x² + y² = 25, y = x + 1