Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circle 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?

$\text{Circles with centre on the line } y = x$ $P y'' + Q y' + 1 = 0$ $\text{Check the four statements about } P, Q$ $(x-a)^2 + (y-a)^2 = r^2 \quad (\text{centre } (a,a) \text{ on } y=x)$ $(x-a) + (y-a)y' = 0$ $1 + (y')^2 + (y-a)y'' = 0$ $y-a = -\frac{1+(y')^2}{y''}, \quad x-a = \frac{(1+(y')^2)y'}{y''}$ $x-y = \frac{(1+(y')^2)(1+y')}{y''}$ $(y-x)y'' + (1+(y')^2)(1+y') = 0$ $(y-x)y'' + (1+y'+(y')^2)y' + 1 = 0$ $P = y - x, \qquad Q = 1 + y' + (y')^2$ $(2): P = y-x \ \checkmark, \qquad (1): P = y+x \ \times$ $(3): P+Q = 1 - x + y + y' + (y')^2 \ \checkmark$ $(4): P-Q = y-x-1-y'-(y')^2 \neq x+y-y'-(y')^2 \ \times$ $\text{Correct statements: } (2) \text{ and } (3)$