# Methods for better understanding a differential equation that can only be solved numerically

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.
• Absolutely. Dynamical systems is the name for the field of mathematics that, among other things, generally concerns itself with the qualitative behavior of systems of differential equations. You might use it, for example, for studying the long-term behavior of systems, determining the conditions for existence and uniqueness of solutions, classifying equilibria, attractors, and other features, or studying how the solution set changes qualitatively with the initial conditions and parameters (which here might be $c$ and the values determining $f$, $g$, and $h$). – Michael Lee May 21 '17 at 6:24