We describe numerical and graphical methods for understanding differential equations.

### Slope fields

We cannot (yet!) solve the differential equation However, from the equation alone, we can describe some facts about the solution.

**slope**of any solution passing through a given point.

Given a differential equation, say , we can pick points in the plane and compute what
the slope of a solution at those points will be. Repeating this process, we can
generate a *slope field*. The slope field for the differential equation looks like this:

Let’s be explicit:

**slope field**, also called a

**direction field**, is a graphical aid for understanding a differential equation, formed by:

- Choosing a grid of points.
- At each point, computing the slope given by the differential equation, using the and -values of the point.
- At each point, drawing a short line segment with that slope.

Here is the slope field for the differential equation , with a few solutions of the differential equation also graphed.

### Autonomous differential equations

Consider the following differential equations

The first differential equation, , is rather easy to solve, we simply integrate both sides.
This type of differential equation is called a **pure-time differential equation**.
Pure-time differential equations express the derivative of the solution explicitly as a
function of an independent variable. We can symbolically describe a pure-time
differential equation as

On the other hand, the second differential equation, does not involve the
independent variable, , at all! Such differential equations are called *autonomous*
differential equations.

**autonomous differential equation**. We can symbolically write to express an autonomous differential equation.

Finally the third differential equation, , expresses as a function of both and the
independent variable . Such a differential equation is called a **nonautonomous
differential equation**. We can symbolically describe a nonautonomous differential
equation as

Since autonomous differential equations **only** depend on the function’s value their
behavior does not depend on the independent variable,

Consider the autonomous differential equation: The constant functions and are
solutions to this differential equation. In fact, for any autonomous differential
equation , where is a function of , if for any constant , then will be a constant
solution to the differential equation. These constant solutions are also known as
**equilibrium** solutions. We can witness these solutions if we inspect the slope field:

Finally, let’s work an example problem:

### Euler’s Method

In science and mathematics, finding exact solutions to differential equations is not always possible. We have already seen that slope fields give us a powerful way to understand the qualitative features of solutions. Sometimes we need a more precise quantitative understanding, meaning we would like numerical approximations of the solutions.

Again, suppose you have set up the following differential equation If we know that solves this differential equation, and , how might we go about approximating ? One idea is to repeatedly use linear approximation.

Let us approximate just using the two subintervals Since , we know that by linear approximation.

**Euler’s Method**.

- Set and .
- Decide either a step-size or how many subintervals you want to divide the interval into. Either way: and .
- Iteratively define for .

Then , and so .

Let’s try an example.