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A curve y = f(x) is such that the perpendicular distance from the origin to the normal at any point P equals the distance of P from the x-axis. Which statement best describes the resulting differential equation/curve?

1. is homogeneous.
2. can be converted into linear differential equation with some suitable substitution.
3. can corresponds to the family of circles touching the x-axis at the origin.
4. can corresponds to the family of circles touching the y-axis at the origin.

Correct answer: can corresponds to the family of circles touching the y-axis at the origin.

Solution

The condition gives 2xy y' = y² - x², whose solution x² + y² = cx is a family of circles centred on the x-axis through the origin, i.e. tangent to the y-axis at the origin (the statement 'is homogeneous' is also true).

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