For example, an electron might travel through a container. It is affected by gravity, by various charges, and by magnetic fields. Gravitational and electrical affects depend on its position, and magnetic affects depend on its velocity. All the affects feed back into its acceleration, which immediately influences its velocity and position. This complicated feedback loop gives us an equation involving a(t), v(t), p(t), and t. Because the equation involves lots of derivatives, we call it a differential equation. To solve the equation is to find a formula for p(t) in terms of t, which is an exact prediction of the electron's future path.

Here's another example: When a chain hangs, the tension on any point on the chain is determined by the overall shape of the whole chain, because points supporting more weight (i.e., higher points) have greater tension. But the curvature of the cable at a point must be inversely proportional to the tension. So the second derivative (curvature), which is of course determined by the function, also determines it via the tension equation. This feedback loop creates a different sort of differential equation. Solving the equation gives the exact shape in which the chain will hang. It turns out to be a catenary.

In each example, the differential equation feels like the exact description of the process itself. Often it comes directly from a general law of physics. Solving the equation tells the implications of the laws, and describes the expected consequences of the rules of the particular system.

Many many different techniques exist to solve differential equations, because many different kinds of differential equations exist. You will learn all the standard techniques, and along the way you will make good use of all of the prerequisites for the course!