Consider the differential equation

.

 Verify that if c is a real constant, then the piecewise function f(x) defined by

 1     if x ≤ c,



                                                           f(x) =        cos(x − c)     if        c < x < c + π,

                                                                           ^ −1                if       x ≥ c + π,

is a solution of the given differential equation.

Choose particular real constants α and β such that f(x), x ∈ [−π,π] is a non-unique solution of the initial value problem

 Explain in the context of your chosen constants α and β why the piecewise function f(x), x ∈ [−π,π] given in (a) is a non-unique solution of the initial value problem in (b).

 State the solution f(x) for each choice of the constant c and sketch in the x,y-plane the non-unique solution curves of the initial value problem in (b) on the interval [−π,π].