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Please, help with DSolve

Alexey Sapozhnikov
Posted 11 years ago
I have to solve the task about mathematical pendulum.
And I have formula:
x''(t)+g/k*sin(x(t))=0, where x - angle of pendulum.
And for solve this we have 
x(t=0) = 0, x'(t=0) = a

But when I write in mathematica:
DSolve[{x''[t] + g/l*Sin[x[t]] == 0, x'[0] == 0, x[0] == a}, x[t], t]
I have three errors:
Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>
DSolve::bvfail: For some branches of the general solution, unable to solve the conditions. >>
DSolve::bvfail: For some branches of the general solution, unable to solve the conditions. >>

I have spend several hours, but with no result...
Please, help me to understand - what happens...
POSTED BY: Alexey Sapozhnikov
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Bill Simpson
Bill Simpson
Posted 11 years ago
This takes what you have already written and makes a few very small changes.
Will this provide the solution that you need?
NDSolve requires that all parameters be given constant numeric values.
In[1]:= g=1; l=1; a=1;
f = x[t]/.NDSolve[{x''[t] + g/l*Sin[x[t]]==0, x'[0]==0, x[0]==a}, x[t], {t, 0, 8}][[1]];
Plot[f, {t, 0, 8}]

Out[2]= ...PlotSnipped...
POSTED BY: Bill Simpson
Thank you very much!
It works!)

But one step further and we stopped again((

Our next step - to find dependence period from starting angle (a).
Period we can find :
g = 1; l = 1; a = 1;
f = x[t] /.
NDSolve[{x''[t] + g/l*Sin[x[t]] == 0, x'[0] == 0, x[0] == a},
  x[t], {t, 0, 12}][[1]]; mt = (t /. FindMinimum[f, {t, 8, 12}][[2]]) - (t /.
    FindMinimum[f, {t, 0, 6}][[2]])
But what I need to do then... I cannot understand...
So I whant to find mt - could you please to help me?

sorry for my english... it is not my organic language an I am under time pressure...
POSTED BY: Alexey Sapozhnikov
May be there are more common way to get period?
POSTED BY: Alexey Sapozhnikov
May be I need to Clear 'a' and to add it as variable in NDSolve?
Something like this: 
g = 1; l = 1;

f = x[t] /. NDSolve[{x''[t] + g/l*Sin[x[t]] == 0, x'[0] == 0, x[0] == a}, x[t], {t, 0, 12}, {a, 0, Pi/2}][[1]];
Plot3D[f, {t, 0, 12}, {a, 0, Pi/2}]
But this code doesn't work emoticon
POSTED BY: Alexey Sapozhnikov
The analytic solution to the pendulum problem involves elliptic integrals:

http://en.wikipedia.org/wiki/Pendulum_%28mathematics%29
POSTED BY: Frank Kampas
DSolve[{x''[t] + Sin[x[t]] == 0, x'[0] == 0, x[0] == 1}, x[t], t]
{{x[t] ->
   2 JacobiAmplitude[
     1/2 (-t Sqrt[2 (1 - Cos[1])] + 2 EllipticF[1/2, Csc[1/2]^2]),
     Csc[1/2]^2]}}
POSTED BY: Frank Kampas
I should have mentioned that my previous post was done on my Raspberry Pi, so it is actually Version 10.  Version 9 returns no result.
POSTED BY: Frank Kampas
We have done it!)
We have to plot dependence between period and starting angle.
And we have done it :
 tt = {};
 
 ti = {};
 Do[f = x[t] /. NDSolve[{x''[t] + g/l*Sin[x[t]] == 0, x'[0] == 0, x[0] == i}, x[t], {t, 0, 12}][[1]];
    m1 = t /. FindMinimum[f, {t, 0, 12}][[2]];
    per = 2*m1;
    AppendTo[tt, per];
    AppendTo[ti, i],
    {i, 0.1, Pi/2 - 0.1, 0.1}]
ListPlot[{tt}, DataRange -> {0.1, Pi/2 - 0.1}]
Thank you very much - without your help this task would be un..resolvable)
POSTED BY: Alexey Sapozhnikov
Rather than running NDSolve multiple times, you can convert the problem into one PDE.
sln = NDSolve[{D[x[t, a], {t, 2}] == -Sin[x[t, a]],
   Derivative[1, 0][x][0, a] == 0, x[0, a] ==  a},
  x, {t, 0, 12}, {a, 0, \[Pi]/2}]
POSTED BY: Frank Kampas
Thank you for this tip Frank, I suspected that there exists more right way).
I doesn't fully understand this single string, but it shows me ways, where I can find gold in future emoticon
POSTED BY: Alexey Sapozhnikov
Another approach, if you're using NDSolve repeatedly, and only changing one or more parameters each time, is to use ParametricNDSolve, to set up the problem.
That avoid repeating the setup each time.
POSTED BY: Frank Kampas
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