# 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?

## 1 Answer

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.

• > "then we could apply standard ODE and dynamical systems theory to make some qualitative statements about the behavior of the system." Could you be a bit more explicit in this step please? (Or point to a a good reference) – rhombidodecahedron May 21 '17 at 5:54
• 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
• Your system in particular could be classified as a "non-autonomous dynamical system," which has its own theory. There are a few books that deal with non-autonomous dynamical systems (I'll come back with some references for you in a bit), but many of the introductory texts work mainly with autonomous systems because there's a rich theory of phase portrait topology for autonomous systems that makes for an easy undergraduate introduction to some areas of contemporary research. – Michael Lee May 21 '17 at 6:26
• Also, just as a continued plug for learning dynamical systems, lots of cool stuff like fractals and chaos theory are basically applications of dynamical systems. – Michael Lee May 21 '17 at 6:33
• Speaking about books, I'd recommend to start with Wiggins and Meiss. They are a bit more down to earth in my opinion, but don't lose the connection with all rigorous theory and have a plenty of examples. – Evgeny May 21 '17 at 6:49