Question

Consider the family of all circles whose centres 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' = dy/dx, y'' = dy/dx), 

which of the following statements (is/are) true? 

(a) P = y + x 

(b) P = y – x 

(c) P + Q = 1 – x + y + y' + (y')² 

(d) P – Q = x + y – y' – (y')²

Answer 1

Answer is (c) P + Q = 1 – x + y + y' + (y')²

Explanation:

Let the family of circles be 

x² + y² – ax – ay + c = 0

Differentiating w.r.t. x, we get

 (y – x)y'' + (1 + y' + y'²)y' + 1 = 0

Thus, 

P = (y – x) and Q = (1 + y' + y'²)

and P + Q = (–x + 1 + y + y' + y'²)