Plot a family of solutions of ODE with singularity

Question

In the post Use Mathematica to plot the flow of an ODE with discontinuity, the following ODE with discontinuous coefficient was solved

T = 1;
Y = ParametricNDSolveValue[{X'[t] == Boole[X[t] > 0], X[0] == x}, X, {t, 0, T}, {x}];
Show[
 Table[
  ParametricPlot[{Y[x][t], t}, {t, 0, T}],
  {x, -1, 1, 0.1}
  ],
 PlotRange -> All,
 AxesLabel -> {"x", "t"}
 ]

I also wish to plot an additional family of solutions displayed in green in the picture below.

• How can a plot just like the one below (but possibly with the t axis being the vertical one) be done starting from the code above?

• Also, is it possible to have the "first" (x=t) and the "last" (x=0) of the green lines displayed in a different color (for example, blue instead of green)?

Answer 1

I'm not sure I understood, but if you only need to add some lines to existing plot from cited question:

T = 10;
Y = ParametricNDSolveValue[{X'[t] == Boole[X[t] > 0], X[0] == x}, 
   X, {t, 0, T}, {x}];
Show[Plot[{x, x + Range[10]}, {x, 0, 10}, 
  PlotRange -> {{-10, 10}, {0, 10}}, PlotStyle -> {Blue, Green}], 
 Table[ParametricPlot[{Y[x][t], t}, {t, 0, T}, PlotStyle -> Red], {x, 
   Complement[Range[-10, 10, 1], {0}]}], 
 Graphics[{Blue, Line[{{0, 0}, {0, 10}}]}], FrameLabel -> {"x", "t"}, 
 Frame -> True, Axes -> False]