Optimizing DE Solutions with Mathematica: Plotting x[t] vs. t"

[Nusc]

DSolve[{x'[t] == (\[Alpha] - (x[t] - 1)^2) x[t], x[0] == 0.004}, x[t],
t]


I get the following message:
Solve::tdep: The equations appear to involve the variables to be solved for in an essentially non-algebraic way. >>

DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution. >>


What do I do?

I need to plot the solution.

 

[CompuChip]

If you just need the plot, you can try solving it numerically.

NDSolve[{x'[t] == (\[Alpha] - (x[t] - 1)^2) x[t], x[0] == 0.004}, x[t], {t, 0, 10}]

(note how you need to specify an interval)

 

[Nusc]

I get the following:

NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.`. >>

 

[CompuChip]

Ah right, my mistake.
You can only solve it numerically if the entire expression is numerical. It chokes on \[Alpha] being symbolic, if you plug in a value for \[Alpha] it does work, e.g.

Block[ { \[Alpha] = 1 },
  NDSolve[{x'[t] == (\[Alpha] - (x[t] - 1)^2) x[t], x[0] == 0.004}, x[t], {t, 0, 10}]
]

Will that do for you, or do you need an exact solution?

 

[Nusc]

I receive the following:

{{x[t] -> \!\(\*
TagBox[
RowBox[{"InterpolatingFunction", "[",
RowBox[{
RowBox[{"{",
RowBox[{"{",
RowBox[{"0.`", ",", "10.`"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}],
False,
Editable->False]\)[t]}}

I can set the value for alpha but I don't obtain a numerical solution. Then I have to plot the solution.

 

[Hepth]

s = NDSolve[{x'[t] == (4 - (x[t] - 1)^2) x[t], x[0] == 0.004}, 
  x[t], {t, 0, 10}]
Plot[Evaluate[x[t] /. s], {t, 0, 10}, PlotRange -> All]

notice you have to put in something for alpha, like previously stated. i did 4.