Mathematics 304 – Ordinary Differential Equations Discussion 1 – Qualitative Techniques for 1st Order ODE August 31, 2000

Background

For a first order ODE \frac{dy}{dx} = f(x, y), the direction field gives valuable qualitative information about solutions. The direction field is a plot in the (x, y)-plane, where a small line segment of slope f(x, y) is placed at the point (x, y). Conceptually, we do this at all points where f(x, y) is defined, but that is impossible to draw, so some finite collection of “representative” points (x, y) is used in practice.

To draw slope fields by hand, a useful technique is:

1. First find the isoclines of the equation – the curves of constant slope value given by the equations f(x, y) = c. (These are the same as the level curves of the slope function f(x, y) as defined in multivariable calculus.)

2. Sketch some collection of the isoclines (often the isoclines for integer slope values, e.g.

c = …, −2, −1, 0, 1, 2, …

are sufficient to see what is happening)

3. Draw in line segments of slope c at points on the isocline for the slope value c.

Then, for instance, we can often get qualitative information about the solutions, since a solution y = ϕ(x) is a function whose graph “follows the direction field” – at each point on the graph, the slope of the tangent line satisfies

ϕ′(x) = \frac{dy}{dx} = f(x, ϕ(x)).

From this picture, the following should be plausible:

Existence and Uniqueness Theorem for ODE Initial Value Problems. Let f(x, y) and \frac{∂f(x,y)}{∂y} be defined and continuous on a rectangle R in the plane containing (x_{0}, y_{0}). Then there is a unique solution ϕ(x) of the IVP

\{ \begin{matrix}\frac{dy}{dx} = f(x, y) \\ y(x_{0}) = y_{0} \end{matrix}

defined on some open interval containing x_{0}.

Today, in our first discussion, we will study direction fields for some differential equations and see what they tell us about solutions.

Discussion Questions

A) Sketch the isoclines with c = −2, −1, 0, 1, 2 for the differential equation

\frac{dy}{dx} = y − x

and use them to sketch the direction field. What can you say about the function ϕ(x) = x + 1 (is it a solution of this equation?) Give rough qualitative sketches of the solutions ϕ(x) with ϕ(0) = 0 and ϕ(−2) = 1. Describe the graphs of solutions of the initial value problems:

\{ \begin{matrix}\frac{dy}{dx} = y − x \\ y(−1) = α \end{matrix}

What happens as x → +∞? How does your answer depend on the value of α?

B) A first order ODE whose slope function is independent of x (that is, which is of the form \frac{dy}{dx} = f(y)) is called an autonomous equation.

1) What do the isoclines of an autonomous equation look like? (What kind(s) of curve(s) are they?)

2) Suppose that f, f′ are continuous on all of R. If x_{0} is an arbitrary x-value, show that the Existence and Uniqueness Theorem for 1st order ODE implies that there is one and only one solution of the IVP

\{ \begin{matrix}\frac{dy}{dx} = f(y) \\ y(x_{0}) = y_{0} \end{matrix}

Explain why this says that the graphs of two different solutions of an autonomous equation never intersect.

3) Suppose that f(y_{0}) = 0 for some particular y_{0}. Explain why these are called equilibrium points for the ODE. What can you say about the constant function ϕ(x) = y_{0}? (Is it a solution of the equation?)

4) For each of the following autonomous ODE, find all the equilibrium points, sketch the part of the direction field for −4 ≤ y ≤ 4 and a “representative sampling” of solutions:

\begin{aligned}\frac{dy}{dx} & = −y^{2} + 5y − 6 = (y − 2)(3 − y) \\ \frac{dy}{dx} & = y^{3} − 2y^{2} + y = y(y − 1)^{2} \\ \frac{dy}{dx} & = y cos(2y) \end{aligned}

5) One important qualitative question about solutions of ODE is: what is their longterm behavior? That is, what is lim_{x→∞} ϕ(x)? In this part, you will study how the graph of the slope function (i.e. the graph w = f(y)) can help determine this. In particular, when is it true that all solutions starting from points near y = y_{0} tend toward y = y_{0} as x → ∞? When is it true that all solutions starting near an equilibrium move away? Is the behavior always the same on both sides of an equilibrium? Use your work from part 4 to help formulate a general statement about how the graph w = f(y) relates to the behavior of solutions near equilibria, and justify your statements.

Assignment

Group write-ups due in class on Tuesday, September 5.