Question: Verify that if c is a constant, then the function defined piecewise by satisfies the differential equation y' = -1- y 2 for all x.

Verify that if c is a constant, then the function defined piecewise by

+1 y(x) = cos(x-c) -1 if x c, ifc

satisfies the differential equation y' = -√1- y² for all x. (Perhaps a preliminary sketch with c = 0 will be helpful.) Sketch a variety of such solution curves. Then determine (in terms of a and b) how many different solutions the initial value problem y' = -√1- y², y(a) = b has.

+1 y(x) = cos(x-c) -1 if x c, ifc

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