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Qualitative Techniques: Slope Fields

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.
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