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(\right. here \textrm{ } y ' = \frac{dy}{dx} , y ' ' = \frac{d^{2} y}{\left(dx\right)^{2}} \left.\right), then which of the following statement(s) is(are) true?

Engineering

Mathematics

Formation of Differential Equation

Question

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' $(here y' = \frac{dy}{dx}, y'' = \frac{d^{2} y}{{dx}^{2}})$ , then which of the following statement(s) is(are) true?

P = y + x

P – Q = x + y – y' – (y')².

P + Q = 1 – x + y + y' + (y')²

P = y – x

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(x – a)² + (y – a)² = r²

2(x – a) + 2(y – a)y' = 0

1 + (y – a) y'' + (y')² = 0

$1 + (y - \frac{x + yy'}{1 + y'}) y'' + {(y')}^{2} = 0$

$1 + y' + (y (1 + y') - (x + yy')) y'' + {(y')}^{2} (1 + y') = 0$

$1 + (y + yy' - x - yy') y'' + y' (1 + y' (1 + y')) = 0$

P = y – x, Q = 1 + y' + (y')²

P + Q = y – x + 1 + y' + (y')²