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 .