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 $g = e^{2u(r)}(dr^2 + r^2\ d\theta^2)$ 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 $r\to\infty$. 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 $ds = e^{u(r)}\ dr$ with $s(0)=0$, and I would have called $K\bigl(h^{-1}(s)\bigr)$ the 'curvature profile'.

Are you sure that you wouldn't have rather had the metric in the form $g = ds^2 + f(s)^2\ d\theta^2$ 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 $f(s) = \sin s$ for $0\le s \le 1$, but on the intervals $1\le s\le 3$ and $3\le s\le 4$, $f$ will be given in terms of translated Airy functions (different ones on the different intervals), and then, for $s\ge 4$, you'll have $f$ be a linear expression in $s$. Of course, determining the constants at the breakpoints so that $f$ is $C^2$ there is probably not going to be doable in any fully explicit fashion.