Mathematics - Differential Equation Question with Solution | TestHub

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 ?