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ODE with in a piece wise plot


Can this be done?

First of all, a basic example and then what I want to change.

E1=If[-1<=x<=0,1,If[3<=x<=5,2,If[-1<=x<=3,(x-1)]]]
Plot[E1,{x,-5,5}]

The purpose of this was to join all three areas into one plot.

Now!  I want to put in a ODE where the (x-1) is.  

Is this possible, and if so, would you mind giving me an example?




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