The first thing to understand from a differential equations class is that most differential equations do not have a closed form solution. DSolve looks for closed form exact solutions, at least if you do not insert any decimal points, and I urge you to not do that because it appears to change many of the rules in hidden and complicated ways.

By changing x'[t]-1-rx[t]-x[t]^2==0 to x'[t]-rx[t]-x[t]*e^x[t]==0 you have replaced an equation which did have a simple closed form solution with one that appears does not, or at least does not have one that could be found in the time I was willing to wait, and I have little patience to wait.

The second thing to understand from a differential equations class is that numerical solutions to differential equations present a wide variety of ways to get incorrect results. You must be very careful to avoid getting meaningless results without knowing it.

When I then try

sols = Table[x /. NDSolve[{x'[t] - r*x[t] - x[t]*e^x[t] == 0, x[0] == 0}, x, {t, 0, 100}][[1, 1]], {r, 1, 3}] ;
Show[Table[ParametricPlot[Evaluate[{sols[[r]][t], D[sols[[r]][t], t]}], {t, 0, 100}], {r, 1, 3}]]

I seem to get an empty plot. When I seem to get an empty plot I ALWAYS FIRST substitute Table for Plot so that I can look at the numbers.

In[2]:= Table[Evaluate[{sols[[1]][t], D[sols[[1]][t], t]}], {t, 0, 100}]

Out[2]= {{0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.}, {0., 0.},
  {0., 0.}, {0., 0.}, {0., 0.}}

When I inspect your DE I realize that x[0]==0 is a valid solution, the plot was actually correct and was just being hidden by the axis.

So it appears that NDSolve did find an approximately correct numerical solution to your DE. Why DSolve did not immediately find this might be a question to think about later.

Hopefully you are learning some of the skills to be able to solve DE using Mathematica. There is an old book that has this as the only subject. You might be able to get an old used inexpensive copy mailed from a bookstore. I have meant to buy a copy and your examples have pushed me into doing this. Thank you.

Hi Bill,

The above code has helped me immensely. Thanks. But when I change the equation, like the following one it does not work.

sols=Table[x/.DSolve[{x'[t]-r*x[t]-x[t]*e^x[t]==0,x[0]==0},x,t][[1,1]],{r,1,3}] Show[Table[ParametricPlot[Evaluate[{sols[[r]][t],D[sols[[r]][t],t]}],{t,0,100}],{r,1,3}]]

regards,

You might want to be cautious with your numerical solution.

Clear[x]; r = 1; eqn = x'[t] - 1 - r*x[t] - x[t]^2 == 0;
sol = x /. NDSolve[{eqn, x[0] == 0}, x, {t, 0, 100}][[1, 1]];

NDSolve::ndsz: At t == 1.2091994106422708`, step size is effectively zero; singularity or stiff system suspected. >>

And

ParametricPlot[Evaluate[{sol[t], D[sol[t], t]}], {t, 0, 100}]

And

Plot[sol[t], {t, 0, 100}]

But

Clear[x]; r = 1; eqn = x'[t] - 1 - r*x[t] - x[t]^2 == 0;
sol = x /. DSolve[{eqn, x[0] == 0}, x, t][[1, 1]];

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

And

ParametricPlot[Evaluate[{sol[t], D[sol[t], t]}], {t, 0, 100}]

And

Plot[sol[t], {t, 0, 100}]

And

In[11]:= sol

Out[11]= Function[{t}, 1/2 (-1 + Sqrt[3] Tan[1/6 (\[Pi] + 3 Sqrt[3] t)])]

Those warning messages are often there for a very good reason.

Even the warning about using Reduce is there for a good reason. Unfortunately it does not include a link to, and I have never been able to track down, a brilliantly well written paragraph in the documentation that describes exactly what that intends you to do.

Your equation is malformed. This variation works:
eqn = x'[t] - 1 - r*x[t] - x[t]^2 == 0