Consider the family of all circles whose centers 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 $y' = \frac{dy}{dx}, \,y'' = \frac{d^{2}y}{dx^{2}}$), then which of the following statements is (are) true ?