Mathematica Solutions to the ISSAC '97 Systems
Challenge
Wolfram Research, Inc.
Problem 10
Consider the following initial value problem.
,
Find the smallest positive number r such that the solution
has a derivative singularity at x = r. Calculate r
to 13 significant digits. Is y(r) infinite or finite? If
y(r)
is finite then compute it to 13 significant digits.
Result
r=1.6443766903388...
y(r)=0.93193876511028...
Method 1: Solve the ODE for and
Let's start with the straightforward numerical solution.
Let's look at the graphs of the function and its derivative. There
seems to be a singularity of on the interval (0,2) at the
point .
If has a singularity at ,
then near this singularity the differential equation reduces to .
The solution of this differential equation has the form .
We conclude that is finite at
and that there is a square root singularity at .
We first solve for up to some point to the left
of the singularity and then switch to the differential equation for .
We can then solve across the singularity for;
it is not a singularity for . We set the appropriate
options in NDSolve to achieve the required precision goal.
Now we change from y[x] to x[y] and continue solving.
To get an approximation to we use the calculated
solution.
We use the resulting interpolating function to find the minimum of
, which is the point .
Here is a plot of the real part of near the square root branch
point. The violet line is the path of integration.
Method 2: Change to the polar coordinate system centered at
{0,1}.
We change variables from ,
to , phi.
This is the new differential equation in .
Here is a plot of the solution.
Now the solution always stays in the order 1, so there is not an
accuracy problem.
The maximum of determines .
Substituting back into the solution of the differential equation
gives .
