A differentiable function--and the solutions to differential equations better be differentiable--has tangent lines at every point. Let's draw small pieces of some of these tangent lines of the function :

**Slope fields** (also called **vector fields** or **direction fields**) are a tool to graphically obtain the solutions to a first order differential equation.
Consider the following example:

The slope, *y*'(*x*), of the solutions *y*(*x*), is determined once we know
the values for *x* and *y* , e.g.,
if *x*=1 and *y*=-1,
then the slope of the solution *y*(*x*) passing
through the point (1,-1) will be
.
If we graph *y*(*x*) in the *x*-*y* plane, it will have slope 2,
given *x*=1 and *y*=-1.
We indicate this graphically by inserting a small line segment at
the point (1,-1) of slope 2.

Thus, the solution of the differential equation with the initial condition *y*(1)=-1 will
look similar to this line segment as long as we stay close to *x*=-1.

Of course, doing this at just one point does not give much information about
the solutions.
We want to do this simultaneously at many points in the *x*-*y* plane.

We can get an idea as to the form of the differential equation's solutions by " connecting the dots." So far, we have graphed little pieces of the tangent lines of our solutions. The " true" solutions should not differ very much from those tangent line pieces!

Let's consider the following differential equation:

Here, the right-hand side of the differential equation depends only on the dependent variable *y*, not on the independent variable *x*. Such a differential equation is called **autonomous**.
Autonomous differential equations are always separable.

Autonomous differential equations have a very special property;
their slope fields are **horizontal-shift-invariant**, i.e. along a horizontal line the slope does not vary.

What is special about the solutions to an autonomous differential equation?

Here is an example of the logistic equation which describes growth with a natural population ceiling:

Note that this equation is also autonomous!

The solutions of this logistic equation have the following form:

As a last example, we consider the non-autonomous differential equation

Now the slope field looks slightly more complicated.

Here is the same slope field again. What is special about the points on the red parabola?

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