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### 0.2Introduction to diﬀerential equations

Note: more than 1 lecture, §1.1 in [EP], chapter 1 in [BD]

#### 0.2.1Diﬀerential equations

The laws of physics are generally written down as diﬀerential equations. Therefore, all of science and engineering use diﬀerential equations to some degree. Understanding diﬀerential equations is essential to understanding almost anything you will study in your science and engineering classes. You can think of mathematics as the language of science, and diﬀerential equations are one of the most important parts of this language as far as science and engineering are concerned. As an analogy, suppose all your classes from now on were given in Swahili. It would be important to ﬁrst learn Swahili, or you would have a very tough time getting a good grade in your classes.

You already saw many diﬀerential equations without perhaps knowing about it. And you even solved simple diﬀerential equations when you took calculus. Let us see an example you may not have seen:

 (1)

Here is the dependent variable and is the independent variable. Equation (1) is a basic example of a diﬀerential equation. In fact, it is an example of a ﬁrst order diﬀerential equation, since it involves only the ﬁrst derivative of the dependent variable. This equation arises from Newton’s law of cooling where the ambient temperature oscillates with time.

#### 0.2.2Solutions of diﬀerential equations

Solving the diﬀerential equation means ﬁnding in terms of . That is, we want to ﬁnd a function of , which we will call , such that when we plug , , and into (1), the equation holds. It is the same idea as it would be for a normal (algebraic) equation of just and . We claim that

is a solution. How do we check? We simply plug into equation (1)! First we need to compute . We ﬁnd that . Now let us compute the left hand side of (1).

Yay! We got precisely the right hand side. But there is more! We claim is also a solution. Let us try,

Again plugging into the left hand side of (1)

And it works yet again!

So there can be many diﬀerent solutions. In fact, for this equation all solutions can be written in the form

for some constant . See Figure 1 for the graph of a few of these solutions. We will see how we ﬁnd these solutions a few lectures from now.

It turns out that solving diﬀerential equations can be quite hard. There is no general method that solves every diﬀerential equation. We will generally focus on how to get exact formulas for solutions of certain diﬀerential equations, but we will also spend a little bit of time on getting approximate solutions.

For most of the course we will look at ordinary diﬀerential equations or ODEs, by which we mean that there is only one independent variable and derivatives are only with respect to this one variable. If there are several independent variables, we will get partial diﬀerential equations or PDEs. We will brieﬂy see these near the end of the course.

Even for ODEs, which are very well understood, it is not a simple question of turning a crank to get answers. It is important to know when it is easy to ﬁnd solutions and how to do so. Although in real applications you will leave much of the actual calculations to computers, you need to understand what they are doing. It is often necessary to simplify or transform your equations into something that a computer can understand and solve. You may need to make certain assumptions and changes in your model to achieve this.

To be a successful engineer or scientist, you will be required to solve problems in your job that you never saw before. It is important to learn problem solving techniques, so that you may apply those techniques to new problems. A common mistake is to expect to learn some prescription for solving all the problems you will encounter in your later career. This course is no exception.

#### 0.2.3Diﬀerential equations in practice

So how do we use diﬀerential equations in science and engineering? First, we have some real world problem we wish to understand. We make some simplifying assumptions and create a mathematical model. That is, we translate the real world situation into a set of diﬀerential equations. Then we apply mathematics to get some sort of a mathematical solution. There is still something left to do. We have to interpret the results. We have to ﬁgure out what the mathematical solution says about the real world problem we started with.

Learning how to formulate the mathematical model and how to interpret the results is what your physics and engineering classes do. In this course we will focus mostly on the mathematical analysis. Sometimes we will work with simple real world examples, so that we have some intuition and motivation about what we are doing.

Let us look at an example of this process. One of the most basic diﬀerential equations is the standard exponential growth model. Let denote the population of some bacteria on a Petri dish. We assume that there is enough food and enough space. Then the rate of growth of bacteria is proportional to the population—a large population grows quicker. Let denote time (say in seconds) and the population. Our model is

for some positive constant .

Example 0.2.1: Suppose there are 100 bacteria at time 0 and 200 bacteria 10 seconds later. How many bacteria will there be 1 minute from time 0 (in 60 seconds)?

First we have to solve the equation. We claim that a solution is given by

where is a constant. Let us try:

And it really is a solution.

OK, so what now? We do not know and we do not know . But we know something. We know , and we also know . Let us plug these conditions in and see what happens.

Therefore, or . So we know that

At one minute, , the population is . See Figure 2.

Let us talk about the interpretation of the results. Does our solution mean that there must be exactly 6400 bacteria on the plate at 60s? No! We made assumptions that might not be true exactly, just approximately. If our assumptions are reasonable, then there will be approximately 6400 bacteria. Also, in real life is a discrete quantity, not a real number. However, our model has no problem saying that for example at 61 seconds, .

Normally, the in is known, and we want to solve the equation for diﬀerent initial conditions. What does that mean? Take for simplicity. Now suppose we want to solve the equation subject to (the initial condition). Then the solution turns out to be (exercise)

We call the general solution, as every solution of the equation can be written in this form for some constant . You will need an initial condition to ﬁnd out what is, in order to ﬁnd the particular solution we are looking for. Generally, when we say “particular solution,” we just mean some solution.

Let us get to what we will call the four fundamental equations. These equations appear very often and it is useful to just memorize what their solutions are. These solutions are reasonably easy to guess by recalling properties of exponentials, sines, and cosines. They are also simple to check, which is something that you should always do. There is no need to wonder if you remembered the solution correctly.

First such equation is,

for some constant . Here is the dependent and the independent variable. The general solution for this equation is

We saw above that this function is a solution, although we used diﬀerent variable names.

Next,

for some constant . The general solution for this equation is

Exercise 0.2.1: Check that the given is really a solution to the equation.

Next, take the second order diﬀerential equation

for some constant . The general solution for this equation is

Note that because we have a second order diﬀerential equation, we have two constants in our general solution.

Exercise 0.2.2: Check that the given is really a solution to the equation.

And ﬁnally, take the second order diﬀerential equation

for some constant . The general solution for this equation is

or

For those that do not know, and are deﬁned by

These functions are sometimes easier to work with than exponentials. They have some nice familiar properties such as , , and (no that is not a typo) and .

Exercise 0.2.3: Check that both forms of the given are really solutions to the equation.

An interesting note about : The graph of is the exact shape a hanging chain will make. This shape is called a catenary. Contrary to popular belief this is not a parabola. If you invert the graph of it is also the ideal arch for supporting its own weight. For example, the gateway arch in Saint Louis is an inverted graph of —if it were just a parabola it might fall down. The formula used in the design is inscribed inside the arch:

#### 0.2.4Exercises

Exercise 0.2.4: Show that is a solution to .

Exercise 0.2.5: Show that is not a solution to .

Exercise 0.2.6: Is a solution to ? Justify.

Exercise 0.2.7: Let . Now try a solution of the form for some (unknown) constant . Is this a solution for some ? If so, ﬁnd all such .

Exercise 0.2.8: Verify that is a solution to . Find to solve for the initial condition .

Exercise 0.2.9: Verify that is a solution to . Find and to solve for the initial conditions and .

Exercise 0.2.10: Find a solution to using your knowledge of derivatives of functions that you know from basic calculus.

Exercise 0.2.11: Solve:

 a) , b) , c) ,  , d) ,  ,

Exercise 0.2.12: Is there a solution to , such that ?

Note: Exercises with numbers 101 and higher have solutions in the back of the book.

Exercise 0.2.101: Show that is a solution to .

Exercise 0.2.102: Is a solution to ? Justify.

Exercise 0.2.103: Let . Try a solution of the form . Is this a solution for some ? If so, ﬁnd all such .

Exercise 0.2.104: Verify that is a solution to . Find and so that satisﬁes and .

Exercise 0.2.105: Solve and .