Study Guide

Differential Equations - Slope Fields

Slope Fields

If you build it, they will come.

Consider the first-order differential equation

We can't solve this d.e. by thinking backwards, but we can still think about how a solution to this d.e. would behave. We know means the slope of y with respect to x. This means any solution to this differential equation is a function y whose slope at any point (x, y) on the function is equal to x + y.

We're going to go back to Leibniz Notation.

Sample Problem

Suppose we have a solution y to the d.e.

that goes through the point (2, 3):

The slope of y at the point (2, 3) must be

Although we don't know what exactly the function y looks like, we do know that at the point (2, 3) the slope of the function (and therefore the slope of its tangent line) is 5:

If we have some other solution y to the d.e.

that goes through the point (4,-1), the slope of y at the point (4,-1) must be

This means the tangent line to y at (4,-1) has slope 3:

Sample Problem

Suppose f is a solution to the d.e.

'(x) = 4xy.

If f passes through the point (3, 10) then the derivative of f at that point is

4(3)(10) = 120.

A graph with lots of little tangent lines, like the one we just drew, is a called a slope field or a vector field. Slope fields are useful for visualizing the solutions to a given differential equation. If you know how the slope of the function behaves, you can see what the overall function looks like.

Because we're just drawing little lines, drawing slope fields is a bit boring and tedious. If we draw the slope field for the d.e.

and include more points, we get something like this:

We're sure you can imagine how awful it would be to need to figure out all those slopes in this picture by hand. Thankfully, this is one place where getting computers to do your work is usually encouraged.

Here's an online tool for drawing slope fields:

You will be asked to match slope fields with their differential equations, or to match differential equations with their slope fields. Our stroll through the slope fields above gave some examples of things you can look for. Here are some questions you can ask yourself when trying to match slope fields and differential equations (you can ask these questions when looking at either a d.e. or a slope field):

  • Where is the slope positive? Negative? Zero?
      
  • What's the slope when x = 0? When y = 0?
      
  • What's the slope when y = x? When y = -x?
      
  • Does the slope depend on both x and y? Just x? Just y?
      
    If the slope only depends on y, then all lines at the same height y will have the same slope:


  • If the slope only depends on x, then all lines at the same x-position will have the same slope:

  • As x approaches ∞ how does the slope change? How about as x approaches -∞? As x approaches 0?


      
  • What happens as y approaches ∞, -∞, or 0?
  • Slope Fields and Solutions

    If we have a slope field for a d.e. and a point in that slope field, we can sketch the solution that goes through that point. The little bits of tangent lines are like arrows telling the function which way to go. If we follow the arrows, we get the solution.

    Sample Problem

    Here's the slope field for the d.e.

    and a point:

    Draw the solution that passes through the point.

    Answer. The little tangent lines trace out the shape of the solution:

    When we use slope fields to sketch solutions to differential equations, the solutions won't necessarily be functions.

    Sample Problem

    Here's the slope field for the d.e.

    and a point:

    This isn't a function because it fails the vertical line test.

    This is cool, because even if we can't get a single-variable function that's a solution to a differential equation, we can still see the shape of the solution on a graph.

    Slope fields are the visual equivalent of IVPs for first-order differential equations. A slope field is the visual equivalent of a differential equation, and a point is the visual equivalent of an initial condition.

    We can draw the solution that goes through a point, given a slope field and a point because there's a Uniqueness Theorem that says such a problem has exactly one answer. 

    Here's the fancy math language, written as bad poetry:

    Given a first-order differential equation and an initial condition
    (so long as the derivative is continuous)
    there is exactly one solution that satisfies both the differential equation and the initial condition.

    Here's the translation for slope fields:

    Given a slope field and a point
    (so long as the derivative is continuous)
    there is exactly one solution that
    goes through that point and has the shape specified by the slope field.

    Since it's very unlikely that you'll ever be given such a problem where the derivative isn't continuous, you can probably get away with thinking that an IVP involving a first-order differential equation has exactly one solution.

  • Equilibrium Solutions

    An equilibrium solution is a solution to a d.e. whose derivative is zero everywhere. On a graph an equilibrium solution looks like a horizontal line.

    Given a slope field, we can find equilibrium solutions by finding everywhere a horizontal line fits into the slope field.

    Equilibrium solutions come in two flavors: stable and unstable. These terms are easiest to understand by looking at slope fields.

    A stable equilibrium solution is one that other solutions are trying to get to. If we pick a point a little bit off the equilibrium in either direction, the solution that goes through that point tries to snuggle up to the equilibrium solution.

    An unstable equilibrium solution is one that the other solutions are trying to get away from. If we pick a point a little bit off the equilibrium, the solution that goes through that point is trying to run away from the equilibrium solution.

    If the solutions are trying to get away on one side and snuggle up on the other side, the equilibrium is still unstable.

    If we're given a differential equation instead of a slope field, we can determine whether each equilibrium solution is stable or unstable by using the differential equation to sketch a very rough slope field.

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