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 $y ^ { \prime } = \frac { d y } { d x } , y ^ { \prime \prime } = \frac { d ^ { 2 } y } { d x ^ { 2 } }$), then which of the following statements is (are) true? (A) $P = y + x$ (B) $P = y - x$ (C) $P + Q = 1 - x + y + y ^ { \prime } + \left( y ^ { \prime } \right) ^ { 2 }$ (D) $P - Q = x + y - y ^ { \prime } - \left( y ^ { \prime } \right) ^ { 2 }$
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 $y ^ { \prime } = \frac { d y } { d x } , y ^ { \prime \prime } = \frac { d ^ { 2 } y } { d x ^ { 2 } }$), then which of the following statements is (are) true?\\
(A) $P = y + x$\\
(B) $P = y - x$\\
(C) $P + Q = 1 - x + y + y ^ { \prime } + \left( y ^ { \prime } \right) ^ { 2 }$\\
(D) $P - Q = x + y - y ^ { \prime } - \left( y ^ { \prime } \right) ^ { 2 }$