Skip to main content

evaluation - Why doesn't NDSolve notice the unknown variable inside the equation?



g = 9.81;
k = 0.009;

r = 10;
b = 0;
ω = Sqrt[g/r];

NDSolve[{r/g*y''[x] + (k*r/g + b*r^2/mg)*y'[x]^2 + Cos[x] - k*Sin[x] == 0,
y[0] == π/2, y'[0] == ω}, y, {x, 0, 2}]

Regarding to this question, NDSolve evaluates the ODE without an error message. NDSolve doesn't evaluate the term b*r^2/mg because b == 0, although the variable mg is unknown. This is not correct in my opinion.


Is it a bug?



Answer




I think the issue behind this question is actually interesting. Not-so-experienced Mathematica users often suffer an illusion: the internal functions will merge when they are used together.


In your case, you probably felt that NDSolve together with the equations and initial conditions have merged to something that will solve the ODE. "So, what happened inside this something?" "Hmm, I don't know, and I believe no body knows, it happened internally." Unfortunately, as said above, it's generally not true. When functions are used together, they will just execute from inner to outer. (This order will be adjusted by attributes like HoldAll, HoldFirst, etc. of course.)


A choice to show the process is to use Trace, you can also try the functions in this post. Here I'll use WReach's traceView2:


NDSolve[{r/g y''[x] + (k r/g + b r^2/mg) y'[x]^2 + Cos[x] - k Sin[x] == 0, 
y[0] == π/2, y'[0] == ω}, y, {x, 0, 2}] // traceView2

enter image description here Pictured by Simon Wood's shadow.


As you see, the equation inside NDSolve is executed before NDSolve begins to work.


Finally, though not quite related, I'd like to mention that, the merge effect does exist in some very rare case, for example this.


Comments

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...