Slope Fields - HartleyMath

In this section, we'll see the first method we have of analyzing differential equations that we don't know how to solve. This method will work for any first-order differential equation. We'll start by learning how to draw a picture that represents all the possible solutions to a first order equation, but we'll quickly see that while it's not that difficult to do, it can be very tedious, so we generally rely on computers to draw these graphs for us. The skill we really need, therefore, is the ability to read these graphs and make sense of them.

Drawing Slope Fields

We'll illustrate this with a simple example: y′=t+yy' = t + yy′=t+y

Clearly, ttt is the independent variable, and yyy is a function of t.t.t. We'll learn in a few sections how to solve this kind of equation, but for now we can't get an explicit solution. However, if we knew the value of yyy for some value of t,t,t, we would also know the value of y′y'y′ at that point. We can therefore find the value of y′y'y′ for every possible combination of ttt and y.y.y. Each combination of ttt and yyy represents a point that we can plot, and y′y'y′at that point is the slope of the solution that would pass through that point.

Therefore we can build a table like the one below: tyy′=t+y000011101112⋮⋮⋮\begin{array}{c c c} t & y & y'=t+y\ \hline 0 & 0 & 0\ 0 & 1 & 1\ 1 & 0 & 1\ 1 & 1 & 2\ \vdots & \vdots & \vdots \end{array}t0011⋮​y0101⋮​y′=t+y0112⋮​​

To draw the slope field, we sketch a short segment at each point with the appropriate slope. The completed graph looks like the following:

What does a slope field mean?

The most basic way to read a slope field is to think of it as a wind map. If you drop a leaf onto this map, where will it go? This of course depends on where you drop it. The slope field represents all the solutions to the differential equation (the family of solutions we saw at the end of the last section). The specific solution depends on the initial condition, which is the spot where the metaphorical leaf is dropped.

For instance, if the initial condition is y(0)=1,y(0)=1,y(0)=1, meaning that y=1y=1y=1 when x=0,x=0,x=0, the solution looks like this curve:

If, on the other hand, the initial condition is y(0)=−2,y(0)=−2,y(0)=−2, the solution looks like this: