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I am no expert on numerical ODEs. But here is what I get. Starting at the right, using the solution of (ODE3) with $U(x) = (c-x)^{1/2}+O((c-x)^{3/2})$ as $x \to c^-$, when I reach $c_o$ it looks like
$$
U(x) = 1831.7 + 0.000103(x-c_o)^{1/2}-13916.8 (x-c_o)+O((x-c_o)^{3/2})
\qquad\text{as } x \to c_o^+
$$$$
U(x) = 1831.7 - 5769.99(x-c_o)^{1/2}-13916.8 (x-c_o)+O((x-c_o)^{3/2})
\qquad\text{as } x \to c_o^+
$$
Then, starting with the soution satisfying
$$
U(x) = 1831.7 - 5769.99(c_o-x)^{1/2}-13916.8 (x-c_o)+O((c_o-x)^{3/2})
\qquad\text{as } x \to c_o^-
$$$$
U(x) = 1831.7 + 0.000103 (c_o-x)^{1/2}-13916.8 (x-c_o)+O((c_o-x)^{3/2})
\qquad\text{as } x \to c_o^-
$$
we end up with $0$ on the $(\log x)^2$ term:
$$
U(x) = O(\;|\log x|\;)
\qquad\text{as } x \to 0^+
$$
The graph seems to be positive everywhere! So we merely need to divide by the integral.
I have Maple working onHere is a plot. (It's slow goingNot very informative.)
Here it is near the point $c_o$:
