I'm not sure that this will help, but let me suggest thinking about the following: You are looking for a metric of the form g=e2u(r)(dr2+r2dtheta2) where u(r) is to be chosen so that the curvature of g is a certain function K(r) and so that u tends to zero as rtoinfty. Now, I wouldn't have called this problem "specifying the curvature profile" just because r won't represent the g-distance from the origin when you are done. Instead, the g-distance s=h(r) from the origin will be given by solving ds=eu(r)dr with s(0)=0, and I would have called Kbigl(h−1(s)bigr) the 'curvature profile'.
Are you sure that you wouldn't have rather had the metric in the form g=ds2+f(s)2dtheta2 where f(0)=0 and f′(0)=1 and then choose f so that it satisfies the equation
f″(s)+K(s)f(s)=0
where K is your given function?
If this is really your problem (and I'm not saying it has to be, but...), then you can, indeed, solve for f explicitly, in a sense, but its definition will be piecewise, of course. You'll have f(s)=sins for 0lesle1, but on the intervals 1lesle3 and 3lesle4, f will be given in terms of translated Airy functions (different ones on the different intervals), and then, for sge4, you'll have f be a linear expression in s. Of course, determining the constants at the breakpoints so that f is C2 there is probably not going to be doable in any fully explicit fashion.
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