Mathematica

[Void123]

I am trying to plot the following numerical solutions:

Input: NDSolve[{y''[x] + (2/x) y'[x] + 1 == 0, y[10^(-10)] == 0.9999999,
y'[10^(-10)] == 0}, y, {x, 0, 30}]

Output: {{y->InterpolatingFunction[{{3.34649*10^-105, 30.}},<>]}}

But when I plot it, I just get a completely blank graph. Why is this?

Does it have something to do with my step size?

[CompuChip]

You should be careful with such results, in this case it gives a list with a list with a function. So the correct plot command would be something like


NDSolve[{y''[x] + (2/x) y'[x] + 1 == 0, y[10^(-10)] == 0.9999999,
y'[10^(-10)] == 0}, y, {x, 0, 30}]
Plot[(y /. Flatten[%])[x], {x, 0, 30}]


Also, I get a warning from NDSolve:
Maximum number of 10000 steps reached at the point x == 3.3464887739011725`*^-105. >>
which means that you probably can't really trust your solution after the point x = 0 :)

You can check this, if you plot the result of the differential equation,
Plot[D[(y /. Flatten[%%])[x], {x, 2}] + 2/x D[(y /. Flatten[%%])[x], x] + 1 /. x -> y, {y, 0, 10}]
If you add PlotRange->All as an option, you see that around 0 you get deviations as large as 0.6

[jackmell]

I am trying to plot the following numerical solutions:

Input: NDSolve[{y''[x] + (2/x) y'[x] + 1 == 0, y[10^(-10)] == 0.9999999,
y'[10^(-10)] == 0}, y, {x, 0, 30}]

Output: {{y->InterpolatingFunction[{{3.34649*10^-105, 30.}},<>]}}

But when I plot it, I just get a completely blank graph. Why is this?

Does it have something to do with my step size?

For starters, you can't plot starting from zero if you run the numerical integrator starting at the point 10^{-10}. Also, the plot function is limited to machine precision which I think is about 16 digits. How about I just do 5?

mysol = NDSolve[{y''[x] + (2/x) y'[x] + 1 == 0,
y[0.00001] == 0.9999999, y'[0.00001] == 0}, y, {x, 0.00001, 30}]
Plot[y[x] /. mysol, {x, 0.00001, 30}]

[Void123]

Thanks guys, I appreciate it.

[Void123]

What if I have different numerical solutions and I wanted to plot them all on the same graph?

Like y''(x) + (2/x)y'(x) + 1 = 0, y''(x) + (2/x)y'(x) + y(x) = 0, etc.?

Thanks.

[jackmell]

What if I have different numerical solutions and I wanted to plot them all on the same graph?

Like y''(x) + (2/x)y'(x) + 1 = 0, y''(x) + (2/x)y'(x) + y(x) = 0, etc.?

Thanks.

Many functions in Mathematica are "listable" meaning you can supply lists as arguments enclosed in curly brackets:

mysol = NDSolve[{y''[x] + (2/x) y'[x] + 1 == 0,
u''[x] + 2/x u'[x] + u[x] == 0, y[0.00001] == 0.9999999,
y'[0.00001] == 0, u[0.00001] == 2, u'[0.00001] == 1}, {y, u}, {x,
0.00001, 30}]
Plot[{y[x], u[x]} /. mysol, {x, 0.00001, 30},
PlotRange -> {{0, 30}, {-20, 20}}]