Mathematics

Problem

Consider the family of all circles whose centres lie on the straight line

y = x

. If this family of circles is represented by the differential equation

Py'' + Qy' + 1 = 0

, where

P, Q

are functions of

x, y

and

y'

(here

y' = \frac{dy}{dx}

y'' = \frac{d^2y}{dx^2}

), then which of the following statements is (are) true? [JEE(Advanced) 2015] (A)

P = y + x

(B)

P = y - x

(C)

P + Q = 1 - x + y + y' + (y')^2

(D)

P - Q = x + y - y' - (y')^2