Calculus Level 4

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 $$y'$$ $$\left( \text{ here } y'=\frac{dy}{dx}, y''=\frac{d^2y}{dx^2} \right)$$ , then which of the following statements is(are) true ?
$\begin{array}{} (1) \, P=y+x \quad \quad \quad \quad & (2) \, P=y-x \\ (3) \, P+Q=1-x+y+y'+(y')^2 & (4) \, P+Q=x+y-y'-(y')^2 \end{array}$
Note :

• Submit your answer as the increasing order of the serial numbers of all the correct options.

• For eg, if your answer is $$(1),(2)$$, then submit 12 as the correct answer, if your answer is $$(2),(3),(4)$$, then submit 234 as the correct answer.

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