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'=dy/dx, y"=d²y/dx²), then which of the following statements is (are) true?

We have ,(x−h)²+(y−h)²=r²
Differentiating and dividing by 2, x−h+yy'−hy'=0 ⇒h=(x+yy')/(1+y')
Again differentiating wrt x, hy''=1+yy''+(y')²
Putting value of h, ((x+yy')/(1+y'))y''=1+yy''+(y')²
(x+yy')y''=(1+yy''+(y')²)(1+y')
xy''+yy'y''=1+yy''+(y')²+y'+yy'y''+(y')³
xy''=1+yy''+(y')²+y'+(y')³
1+yy''+(y')²+y'³+yy''−xy''=0
1+y'(1+y'+(y')²)+y''(y−x)=0
⇒P=y−x,Q=1+y'+(y')²
⇒P+Q=1−x+y+y'+(y')²