Introduction to Differential Equations

Modeling the world around us is an important application of mathematics. Many models involve what are called differential equations. These are equations that relate derivatives to one another. We can measure the rate at which something is changing with respect to time or space and if we know something about the relationship between these rates, we can use this information to create models of reality. For example, to model an earthquake, we want to describe the motion of a wave through space and time. We may not know what the mathematical equation that describes the wave is, but we can measure the rate of change of the displacement caused by this wave as it moves through space and time. Using this information can relate the rate of change in time to the rate of change in space of the displacement caused by a wave moving through the earth. This information allows us to model an earthquake without having direct access to an equation that describes the earthquake.

This kind of equation, where you find a relationship between rates of change, is called a differential equation. Differential equations often involve multivariable functions, and are a topic covered in MATH 256. However, today we will get a preview of this topic by using integration to solve a differential equation!

We will be exploring a model of a physical system. Imagine you have a metal rod and you are heating the rod from below with some kind of heat source, say a candle.

1. We will begin by thinking about what is happening in this physical system.

(a) What would you expect to happen to the rod as time goes on?

(b) What would you expect to happen to the rod as time goes on, if you remove the heat source?

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2. From here on out we will assume that the heat source is left under the rod and we will assume that the system is in equilibrium. That is, the heat coming into the rod from the heat source is equal to the heat leaving the rod. Let y(t, x) be an equation that describes the heat in the system at time t and location x along our metal rod. Because of our equilibrium assumption, we can drop the time component and say that y is just a function of x.

We can model this system with an equation that relates the rate of change of the heat of the system

(rod) with respect time ∂y

∂t to the second derivative of heat with respect to space plus a function that

describes how heat is being added to the system. That is

∂y

∂t =

∂2y

∂x2 + (sin(x)x− 1) .

Since we are in equilibrium, ∂y

∂t = 0 and we get:

∂2y

∂x2 = − sin(x)x+ 1.

(a) Using reasoning about rates of change, explain why the equilibrium assumption leads to ∂y

∂t = 0.

(b) Using the Fundamental Theorem of Calculus, find the derivative of y(x).

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(c) Using your work in part (a) find the general form of y(x). (Hint: you can combine the constants that arose from the integration in part (a) on one side of the equation and consolidate them into a new constant that we can call b.)

(d) Suppose that y(0) = 0 and y(π) = 2, using this information and your work in part (b) and (c) , what is y(x)?

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