Verify that if c is a constant, then the function defined piecewise bysatisfies the differ...

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

satisfies the differential equation 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(a) = b has.