I was up till 3:30. There was no way that I would sleep before the problem would.

As it turns out, the integral equation that I discussed earlier is well-known. It is an example of a linear Volterra equation of the first kind. \[f(c)=\int_{a}^{c}{K(c,x)\rho(x)\textrm{ d}x}\] The function \(K\) is known as the kernel. In our case, the kernel is: \[K(c,x)=c+b-x\] The idea here is to use the Leibniz integral rule (which follows from Fundamental Theorem). By differentiating both sides with respect to \(c\), we obtain: \[\frac{\textrm{d}}{\textrm{d}c}\left(\int_{c}^{\infty}{(c+b-x)\rho(x)\textrm{ d}x}\right)=\frac{\textrm{d}R}{\textrm{d}t}\lim_{R\rightarrow\infty}{\left[(c+b-R)\rho(R)\right]}-b\rho(x)+\int_{c}^{\infty}{\rho(x)\textrm{ d}x}\] Observe that if \(G(c, R)=\int_{c}^{R}{(c+b-x)\rho(x)\textrm{ d}x}\) is well-behaved, then we may interchange the limit and derivative above to obtain: \[\frac{\textrm{d}R}{\textrm{d}t}\lim_{R\rightarrow\infty}{\left[(c+b-R)\rho(R)\right]}=\lim_{R\rightarrow\infty}{\left[(c+b-R)\rho(R)\frac{\textrm{d}R}{\textrm{d}t}\right]}=0\] As cited, the conditions we want for \(G\) for this manipulation to be valid are:

  • The derivative of \(G\) WRT \(c\) must converge uniformly,
  • \(G\) itself must converge for some \(c\).

Both of these conditions are intuitively obvious. By our original Volterra equation, the second condition is satisfied \(\forall c>0\). We assume the first condition now, and can prove it later by showing that the product of the kernel and \(\rho\) is uniformly convergent.

Anyway, we have: \[b\rho(c)=\int_{c}^{\infty}{\rho(x)\textrm{ d}x}\] Let \(P’=\rho\). Then, by the Fundamental Theorem: \[bP’(c)=\lim_{R\rightarrow\infty}{P(R)}-P(c)\] Let \(L=\lim_{R\rightarrow\infty}{P(R)}\). Then, the equation above rearranges to: \[P’+\frac{1}{b}P=\frac{L}{b}\] This is simply a first-order linear ODE. We solve this by using an integrating factor of \(\exp{\int{\frac{1}{b}\textrm{ d}{x}}}\). This yields: \[P(x)=L+Ce^{-x/b}\] Differentiating this, we obtain: \[\boxed{\rho(x)=Ce^{-x/b}}\] Obviously with \(C>0\).

Uniform convergence is forced by the fact that this solution satisfies our Volterra equation. It can probably also be proven with \(\epsilon\)-\(\delta\) calculations. That’s for another time. And I’m done with mechanics for this weekend I think.