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From: Sam Takoy on 28 Jul 2010 02:54
Hi,

This:

NDSolve[{ 1 + r'[x]^2 - r[x] r''[x] == 0, r[0] == r[1] == 1}, r, {x, 0, 1}]

should solve the catenary problem, but it gives 1/0 errors. Can you
suggest a fix?

Many thanks!

Sam

From: Kevin J. McCann on 29 Jul 2010 06:44
Your problem is likely in the attempt to solve a boundary value problem
rather than an initial value problem. Here is a quick shooting method
that gets the answer:

(* Define the shooter. Takes an initial slope of a and returns r[1] *)
Clear[shoot, a]
shoot[a_?NumberQ] := Module[{r, x, y},
y = r /. NDSolve[{1 +
\!\(\*SuperscriptBox["r", "\[Prime]",
MultilineFunction->None]\)[x]^2 - r[x]
\!\(\*SuperscriptBox["r", "\[Prime]\[Prime]",
MultilineFunction->None]\)[x] == 0, r[0] == 1, r'[0] == a},
r, {x, 0, 1}][[1]];
y[1]
]

(* Find the correct initial slope *)

\[Alpha] = a /. FindRoot[shoot[a] == 1, {a, -1, 1}]

(* Now that we have the correct slope, solve the DE *)

R = r /. NDSolve[{1 +
\!\(\*SuperscriptBox["r", "\[Prime]",
MultilineFunction->None]\)[x]^2 - r[x]
\!\(\*SuperscriptBox["r", "\[Prime]\[Prime]",
MultilineFunction->None]\)[x] == 0, r[0] == 1, r'[0] == \[Alpha]},
r, {x, 0, 1}][[1]]

(* Plot it *)

Plot[R[x], {x, 0, 1}, PlotRange -> {0.5, 1}]


Kevin

Sam Takoy wrote:
> Hi,
>
> This:
>
> NDSolve[{ 1 + r'[x]^2 - r[x] r''[x] == 0, r[0] == r[1] == 1}, r, {x, 0, 1}]
>
> should solve the catenary problem, but it gives 1/0 errors. Can you
> suggest a fix?
>
> Many thanks!
>
> Sam
>

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