I have a differential equation of the following form:
$$ \frac{\text{d}}{\text{d}x} y(x) = f(x) - cx^2g(x) \int_{t=x}^1 \frac{h(t)}{t^2}y(t) \; \text{d}t $$
where $c$ is a constant and $f$, $g$, and $h$ are functions of $x$ that are known numerically at points $0=x_1, x_2, \ldots x_n=1$.
I can solve this for $y(x)$ under assumption of some boundary conditions. No problem. But, I would like to understand this equation better. What are some good options for studying and visualizing this equation?
You might consider rewriting this using an extra function $z(x) = \int_x^1 \frac{h(t)y(t)}{t^2}\,\mathrm{d}t$. Then, we have the coupled equations \begin{align*} y'(x) &= f(x)-cx^2g(x)z(x) \\ z'(x) &= \frac{h(x)y(x)}{x^2} \end{align*} with an added initial condition on $z$ to enforce our ad hoc definition. We might then choose some kind of interpolation for $f$, $g$, and $h$, and then we could apply standard ODE and dynamical systems theory to make some qualitative statements about the behavior of the system.