I am modeling a cyclic voltammetry experiment. The mathematical model for it boils down to a Volterra integral equation of the first kind, with an unknown function $\chi(z)$ and known constants $\xi$ and $\theta$:
$$\int_0^{\sigma t}\frac{\chi(z)\mathrm{d}z}{(\sigma t-z)^{1/2}} = \frac{1}{1+\xi\theta e^{-\sigma t}}.$$
Is this a well-posed integral equation? I am troubled by the fact that the right-hand side does not vanish as $\sigma t$ tends to zero.
To solve this integral equation, I am differentiating both sides with respect to $\sigma t$ to obtain an ordinary differential equation:
$$\frac{\chi(z)}{\sqrt{z}}+2\sqrt{z}\chi^\prime(z) = \frac{-\xi\theta e^{-z}}{(1+\xi\theta e^{-z})^2}.$$
What boundary condition should I assume to solve this differential equation?