Mathematics 134 – Intensive Calculus for Science 2 Discussion 5 – Differential Equations and Slope Fields April 4, 2001

Background

In class on Tuesday, we introduced the idea of a differential equation – a relation between derivatives of an unknown function, that function, and other known functions of one variable. The order of a differential equation is the highest order derivative of the unknown function that appears. For instance,

\frac{d^{2}y}{dt^{2}} + 5\frac{dy}{dt} + 6y = cos(4t)

is a second order differential equation for the unknown function y. Today we want to practice recognizing when a function solves a differential equation.

We will also introduce a graphical way to understand how the graphs of solutions of a general first order differential equation \frac{dy}{dx} = f(x, y) are determined by the equation. This will use what is called the slope field of the equation. The idea is the following. If we have any function y = y(x), then the value of the derivative \frac{dy}{dx} tells us the slope of the graph (i.e. the slope of the tangent line to the graph) at each point. Hence the differential equation \frac{dy}{dx} = f(x, y) says that in order for a function y = y(x) to be a solution, at each point (x_{0}, y_{0}) = (x_{0}, y(x_{0})) on the graph of the solution function, the slope of the tangent line at that point has to be equal to the value we get from the “slope function” f at that point: f(x_{0}, y_{0}).

Hence, to visualize the graphs of solutions, we can make a plot showing the slopes f(x_{0}, y_{0}) at a collection of points. (We usually use points in some grid or subdivision of a part of the xy-plane.) The slope value at (x_{0}, y_{0}) is indicated by putting in a small line segment of that slope with its midpoint at (x_{0}, y_{0}). Then the graphs of solutions can be visualized by drawing in curves that “follow” the slope field – that are tangent to the slope field line segments at each point they pass through.

Slope field for \frac{dy}{dx} = y cos(\sqrt{x} sin(x))

and solution with y(1) = 3 Discussion Questions A) Which of the following list of functions are solutions of which of the list of differential equations. (There may be zero, one, or more than one solution for each of the differential equations.) Functions:

y_{1}(x) = sin(4x) y_{2}(x) = cos(4x) y_{3}(x) = e^{−4x} y_{4}(x) = x^{4} y_{5}(x) = x^{−3} Differential Equations:

1) \frac{d^{2}y}{dx^{2}} = −16y

2) \frac{dy}{dx} = −4y

3) \frac{dy}{dx} = 4y

4) x^{2} \frac{d^{2}y}{dx^{2}} − 12y = 0 B) Problem 1 from Section 9.2 in our text C) Problem 7 from Section 9.2 in our text Assignment

Group writeups due in class on Tuesday, April 10.