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

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

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

Blog

Search This Blog