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### Geometric interpretation of a differential equation

#### Slope Fields

Consider a differential equation $\displaystyle\frac{dy}{dx}=f(x,y)$. Since the derivative is the slope of the tangent line, we interpret this equation geometrically to mean that at any point $(x,y)$ in the plane, the tangent line must have slope $f(x,y).$ We illustrate this with a slope field, a graph where we draw an arrow indicating the slope at a grid of points. The slope field for $\displaystyle\frac{dy}{dx}=x+y+2$ is illustrated at the right. |

The solution to a differential equation is a curve that is tangent to the arrows of the slope field. Since differential equations are solved by integrating, we call such a curve an integral curve. This picture illustrates some of the integral curves for $\displaystyle\frac{dy}{dx}=x+y+2$. You can see there are a lot of possible integral curves, infinitely many in fact. This corresponds to the fact that there are infinitely many solutions to a typical differential equation. To specify a particular integral curve, you must specify a point on the curve. Once you specify one specific point, the rest of the curve is determined by following the arrows. This corresponds to finding a particular solution by specifying an initial value. |

In this lab, you will experiment with the slope fields and integral
curves
for a variety of different equations. The goal is to get a geometric
concept of what a differential equation means, to go along with the
algebraic techniques to solve such equations. Launch the
slope field app.
This brings up an app where you can type in differential equations and see
their slope
fields and integral curves. When you type in the differential equation,
use ^ for powers, and functions (such as
sin(x), exp(x), log(x), etc.) must use parentheses. Once you type in a
formula, you should press enter or click the `"y(-4) ="`

*Write up your answers to the following 8 questions neatly on a separate
sheet of paper. Be sure to use complete sentences in your answers to
problems 3-6.*

The first pair of problems just ask you to play around with a couple of slope fields and initial conditions to get started and to make sure you understand the connection between the initial condition $y(-4) = y_0$ and the geometric picture of the integral curve.

- Let $y'=\cos(x)+\sin(y)$. Find a value for
y(-4) so that-1 < y(5) < 1 . - Let $y'=y-x/2$. Find a value for
y(-4) so that-3 < y(5) < 0 .The next problem illustrates that the solution to a differential equation may not be defined over the whole real line, but may only exist in some particular interval.

- Let $y'=-(x+2)/y$. Draw 33 integral curves. Note that none of the
curves drawn extend entirely across the graph. You may have observed
earlier that if you point at one of the integral curves with the mouse,
that curve turns purple (and if you have multiple integral curves that get
asymptotically close, the mouse is pointing at all of them and they all
turn purple, which can lead to some pretty effects). Point at one of the
integral curves. Note that only half the circle turns purple. Write a two
or three sentence explanation for (i) why the solutions don't continue all
the way across the graph and (ii) why the integral curve isn't the full
circle.
A theorem we will discuss later says that for a "reasonable" initial value problem, there will be one and only one solution, which will be defined on some interval about the initial point. That there is a solution should seem reasonable at this point, just start at the initial value and follow the slope field and you will get the graph of the solution. The last example should illustrate why you might get a solution defined only in an interval.

- In solving the first two problems, you almost certainly observed
and used (if only implicitly) the fact that solution curves didn't cross
to narrow the range of possible initial values to try. Explain how this
fact implies that a "reasonable" initial value problem can't have two
solutions.
In the final 4 problems, you will develop some feeling for the connections between the slope field and certain types of differential equations. First you will consider some examples, and then you can use the ideas you've developed to try to recognize a couple of equations by their slope fields.

- Look at several examples of slope fields of the form $y'=f(x)$ where
the right-hand-side doesn't depend on y. Press the
"Draw 33 Int. Curves" button to look at the integral curves for these slope fields. You should observe a pattern in such fields. Write one or two sentences to describe how you can tell if a slope field corresponds to such an equation. - Look at several examples of slope fields of the form $y'=f(y)$ where
the right-hand-side doesn't depend on x (such equations are called
"autonomous"). You should observe a pattern in such fields. Press the
"Draw 33 Int. Curves" button to look at the integral curves for these slope fields. Do the integral curves have the same sort of pattern as for the fields considered in problem 5? Write one or two sentences to describe how you can tell if a slope field corresponds to such an equation. At the right is the slope field for $y' = f(x,y)$. Based on the graph, identify where the function $f(x,y)$ is positive, negative, and zero. Is $f(x,y)$ a function of $x$ alone, $y$ alone, or a function of both variables together? Find a function $f(x,y)$ whose slope field looks like this. At the right is the slope field for $y' = g(x,y)$. Where is the function $g(x,y)$ positive, negative, and zero? Is $g(x,y)$ a function of $x$ alone, $y$ alone, or a function of both variables together? Find a function $g(x,y)$ whose slope field looks like this.

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010 Andrew G. Bennett