# singularity in NDSolve for some values of parameter

my equation is:

k[a_?NumericQ, b_?NumericQ, c_?NumericQ] := NDSolve[{r^3 b x''[r] y[r] - 12 x[r] y[r] -
4 r^3 b y''[r] x[r] == 0, 4 (2 x'[r]^2 +
2 x[r]x''[r])/Sqrt[c] + (2 y'[r]^2 +
2 y[r] y''[r])/Sqrt[c] == -4b/r,x[1] == 0, y[1] == Sqrt[-4 b Log[2/b]/(c^(-1/2))],
y'[1] == c^(1/4) (-4 b - 4 b Log[2 /b])/(2 Sqrt[-4 b Log[2/b] ] ), x'[1] == a},
{x, y}, {r, 1, b/2}, MaxSteps -> 1000000]


and then I plot it:

Plot[Evaluate[{x[r], y[r]} /. k[1, 50, 10^(-7)]], {r, 1, 25}]


this one is okay, cause x[r] and y[r] both equal to 0 at r=b/2

if I set a=0 and the other parameters stay the same

I do not know how comes this sigularity at r=b/2, basing on the second equation in NDSolve, 4 (2 x'[r]^2 + 2 x[r]x''[r])/Sqrt[c] + (2 y'[r]^2 + 2 y[r] y''[r])/Sqrt[c] == -4b/r comes from the second derivative of r in this equation:

c^(-1/2) 4 x[r]^2 + c^(-1/2) y[r]^2 == -4 r b Log[(2 r)/b]

so it's easy to check, if x[r]=0 all the way, y[r] should be 0 at r=b/2

does anyone know how to resolve the infinity at r=b/2 for any values of the parameter? a>=0, b>2, 10^(-3)>c>10^(-7)

thanks a lot

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I think this is just another story of WorkingPrecision, let's try a higher one, say, 32:

k[a_?NumericQ, b_?NumericQ, c_?NumericQ] :=
NDSolve[{r^3 b x''[r] y[r] - 12 x[r] y[r] - 4 r^3 b y''[r] x[r] == 0,
4 (2 x'[r]^2 + 2 x[r] x''[r])/
Sqrt[c] + (2 y'[r]^2 + 2 y[r] y''[r])/Sqrt[c] == -4 b/r,
x[1] == 0, y[1] == Sqrt[-4 b Log[2/b]/(c^(-1/2))],
y'[1] == c^(1/4) (-4 b - 4 b Log[2/b])/(2 Sqrt[-4 b Log[2/b]]),
x'[1] == a}, {x, y}, {r, 1, b/2}, WorkingPrecision -> 32]

Plot[Evaluate[{x[r], y[r]} /. k[0, 50, 10^(-7)]], {r, 1, 25}]


OK, the singularity disappears.

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yes, it works. can you explain more to me why this works? thanks – 3c. Jul 26 '13 at 14:07
@3c. Well, to be honest, I'm not able to tell the exact reason, but based on my experience, if the plot of a function becomes unusual(singularity, hard oscillation etc while the function should not possess these features) in a point or in a small interval, it's often related to a WorkingPrecision that isn't high enough. There're numbers of posts about WorkingPrecision in this site, e.g. this, you can have a search for more details. – xzczd Jul 26 '13 at 14:26
thanks very much – 3c. Jul 26 '13 at 14:36