Question

1. Find the differential equation for X2+y2=r2 , where r is an arbitrary constant​

Answer 1

Answer:

Step-by-step explConsider the circle x2+y2=r2. Note that y′=−x/y, and furthermore, for a given value of y, there are two possible values of x, but only one up to negation, and thus, two possible values of y′, but only one up to negation. This means there is a unique possible value of y′2 for any given y; specifically, y′2=x2y2=r2−y2y2=(ry)2−1.

This final equation y′2=(ry)2−1 no longer mentions x at all, and thus works for centers anywhere along the x axis. It may be considered as having many “non-smooth solutions”, so to speak, which switch at times between the two possible circles a given point could be on, but its smooth solutions are indeed precisely the circles of fixed radii and centers on the x axis.

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