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Consider all circles whose centres lie on the line y = x. If this family satisfies a differential equation P y'' + Q y' + 1 = 0 (P, Q being functions of x, y, y'), which of the following is correct?
- P = y + x
- P = y - x
- P + Q = 1 - x + y + y' + (y')²
- P - Q = x + y - y' - (y')²
Correct answer: P = y - x
Solution
Eliminating the centre and radius gives P = y - x and Q = 1 + y' + y'², so P = y - x is true (and P + Q = 1 - x + y + y' + (y')² also holds).
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