Plotting boundary value expression against a parameter

I am very new to Mathematica and have a question regarding plotting a ODE boundary value problem.

Let say I have an ODE

alpha = 1/2;
beta = -5;
eq = beta*f'''[x] + alpha*f[x]*f'[x] == 0
bc = {f[0] == 0, f'[0] == 0, f'[10] == 1};
NDSolve[{eq, bc}, f[x], {x, 0, 10}]


I want to plot alpha*f''[0] + (f''[0])^3 vs beta. In the plot, beta should take the range [-5, 5]. NDSolve should use shooting method for the numerical solution.

Also, how I can extract data from that plot as a .txt file.

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Could you try to write down some Mathematica code? – Dr. belisarius Jul 22 '13 at 4:27
No idea how to write mathematica code here. Sorry mate – MMM Jul 22 '13 at 4:34
you can take infinity to be some thing like 10 or 20 – MMM Jul 22 '13 at 4:40
Thanks mate. Now it looks very clear. Cheers – MMM Jul 22 '13 at 4:46

This is not a complete solution as some issues remain. but may be will help you get started, as I have to go.

There is some issues when beta is takes some values (between -1 and 1). NDSolve complains for some values of beta.

Failed to converge to the requested accuracy or precision \
within 100 iterations.


This generates the plot you wanted, but you might have to play with NDSolve options or make sure your parameters are ok.

y[beta_] :=
Module[{eq, sol, x, f, bc, max = 10, alpha = 0.5, r},
eq = beta*f'''[x] + alpha*f[x]*f'[x] == 0;
bc = {f[0] == 0, f'[0] == 0, f'[max] == 1};
sol = First@NDSolve[{eq, bc}, {f[x], f'[x], f''[x]}, {x, 0, max}];
r = f''[x] /. sol /. x -> 0;
alpha*r + beta*r^3];
data = Table[{beta, y[beta]}, {beta, -5, 5, 1}];
ListPlot[data, Joined -> True, Mesh -> True, Frame -> True,
FrameLabel -> {{"y(beta)", None}, {beta, "y[beta] vs. beta"}}]


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data = Table[{lambda, y[lambda]}, {lambda, -5, 0, 1}]; – MMM Jul 22 '13 at 5:48
Trying to plot alphar + betar^3 vs lambda here. Can we use shooting method? – MMM Jul 22 '13 at 5:49
I wanted the answer you already give me but I am unable to execute the code. STill I got an error. Could you please upload ur .nb file? As for as the change is concern, I was hope to give an ode with which NDsolve may have no issue. – MMM Jul 22 '13 at 6:03
– MMM Jul 22 '13 at 6:09