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.
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:
Suppose f is a solution to the d.e.
f '(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?