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 $P y^{\prime \prime}+Q y^{\prime}+1=0$, where $P, Q$ are functions of $x, y$ and $y^{\prime}$ (here $\left.y^{\prime}=\frac{d y}{d x}, y^{\prime \prime}=\frac{d^2 y}{d x^2}\right)$, then which of the following statements is (are) true?