Consider the family of all circles whose centers lie on the straight line $\mathrm{y}=\mathrm{x}.$ If this family of circles is represented by the differential equation $\mathrm{P}\mathrm{y}"\mathrm{ }+\mathrm{ }\mathrm{Q}\mathrm{y}\mathrm{\prime }\mathrm{ }+\mathrm{ }1\mathrm{ }=\mathrm{ }0,\mathrm{ }$ where $\mathrm{P},\mathrm{ }\mathrm{Q}$ are functions of $\mathrm{x},\mathrm{ }\mathrm{y}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{y}\mathrm{\prime }$ $\left(\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{ }{\mathrm{y}}^{\mathrm{\prime }}=\mathrm{ }\frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{x}},\mathrm{ }\mathrm{y}"\mathrm{ }=\mathrm{ }\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{x}}^2}\right)$ , then which of the following statements is (are) true ?

• A

  $\mathrm{P}=\mathrm{y}+\mathrm{x}$
• Correct answer
  B

  $\mathrm{P}=\mathrm{y}-\mathrm{x}$
• Correct answer
  C

  $\mathrm{P}+\mathrm{Q}=1-\mathrm{x}+\mathrm{y}+{\mathrm{y}}^{\mathrm{\prime }}+{\left({\mathrm{y}}^{\mathrm{\prime }}\right)}^2$
• D

  $\mathrm{P}-\mathrm{Q}=\mathrm{x}+\mathrm{y}-{\mathrm{y}}^{\mathrm{\prime }}-{\left({\mathrm{y}}^{\mathrm{\prime }}\right)}^2$

Hence, the correct answer is options: 'B', 'C'.