Kansas State University

"Mathematics is the language in which God has written the universe." - Galileo

Differential equations are one of the most important tools in writing such descriptions. A differential equation is just an equation involving one or more derivatives. Differential equations are important because they let you write a description of a situation in terms of the forces acting in the situation. Consider Newton's second law of motion $$\text{Force}=\frac{dp}{dt}$$ where $p$ is momentum and $t$ is time. This is probably more familiar to you as $F=ma$, where we have made the assumption that mass is constant. Suppose you are told the acceleration of gravity on the moon is $1.6 \text{m}/\text{sec}^2$ . If you drop a mass above the surface of the moon, how far will it fall in 1 second? Here you know acceleration, but what you want to know is distance. This leads to the initial value problem find $x(1)$ where $d^2x/dt^2 = -1.6$, $x(0) = 0$, $x'(0)=0$. In a physical situation you often know the forces acting on an object and then have to determine the future behavior of the object. This will lead to a differential equation. Of course differential equations occur in many other subjects besides physics. The idea of thinking of a situation in terms of the "forces" acting on it makes sense not just in physics but in economics, population biology, and many other disciplines. We will meet a variety of applications for differential equations during this course.

While the example above leads to a specific equation that we need to solve, there are many situations where the key is being able to interpret the consequences of a differential equation without actually finding a solution. For example, Newton's third law says that for every action (force) there is an equal and opposite reaction. This says the total net force in every situation is 0 (though there may be non-zero forces acting on any particular object in the situation). Now since $\text{Force}=\displaystyle\frac{dp}{dt}$, this says $\displaystyle\frac{dp}{dt}=0$, i.e. total momentum has a 0 derivative, so momentum is conserved. Here we have deduced a key physical law by interpreting what the differential equation is saying, rather than solving a specific problem. In this class you will be asked both to solve problems and also to use and interpret differential equations to describe and explain mathematical and scientific ideas.

Mathematical language can be approached in several different ways. You can look at problems both algebraically and graphically. This class will push you to translate between these different approaches so you can better understand what each tells you and be better prepared to handle real problems where you may need to switch back and forth between different representations. In the rest of this section, we will introduce some definitions and notation, concentrating on the algebraic viewpoint. We will discuss the geometric viewpoint in the next section (which we will cover in lab).

Definitions

EXAMPLES: $$ \begin{align} \frac{dy}{dx}&=2x^2+3y \\ \frac{d^2y}{dx^2}+2x\frac{dy}{dx}+(x^2+1)y&=0 \\ \frac{dp}{dt}&=-.01p^2+p \\ x''+2x'+10x&=10\cos(t) \\ \exp(f(x))\sqrt{1+(f'(x))^2} &= \frac{\exp(f(x))\left((f'(x))^2+f''(x)\right) (1+(f'(x))^2) - \exp(f(x))(f'(x))^2f''(x)} {\left(1+(f'(x))^2\right)^{3/2}} \end{align} $$ In the first example, $y$ is to be a function of $x$ whose derivative is $2x^2$ plus 3 times the original function $y$. Note that we write $y$ instead of $y(x)$. We will usually not distinguish between functions and variables in our notation. It will actually make things easier in some problems to just think of functions as variables and treat them accordingly. Of course, in some situations you may want to use the $f'$ notation rather than $df/dx$ and the names of the variables can be whatever is convenient. The key is having an equation involving derivatives, not what notation is used.

The order of a differential equation is the highest order of derivative that occurs in the equation. The first and third examples are first order while the the other examples are second order. A particular solution to a differential equation is a function that satisfies the equation.

EXAMPLES: $$ \begin{align} y(x) = x^2 &\text{ is a particular solution to } \frac{dy}{dx}=2x \\ y(x)=x^2+1 &\text{ is a particular solution to } \frac{dy}{dx}=2x \end{align} $$ This illustrates a problem with differential equations, solutions are usually not unique (a problem has a "unique" solution if it has only one solution). There are two common approaches to resolving this difficulty. One is to ask for the general solution instead of the particular solution. A general solution is a set of solutions to a differential equation with as many arbitrary constants as the order of the equation.

EXAMPLE: $$ y(x)=x^2+C \text{ is the general solution to } \frac{dy}{dx}=2x $$ For any choice of the arbitrary constant $C$ we get a solution to the equation. The two solutions in the previous example come from choosing $C=0$ and $C=1$. The equation is first order so we only expect one arbitrary constant. In many cases the general solution gives all the solutions but this need not be the case. A singular solution is a particular solution to an equation which is not an instance of the general solution. We will see examples of this later. (The term singular solution is sometimes given a more restrictive and technical meaning in advanced courses.)

A second approach to dealing with the non-uniqueness of solutions is to narrow the problem. We ask not for any old solution to the problem but rather for a solution which also has some additional properties. One common situation is an initial value problem where one wants the solution to a differential equation which takes a specified value at a given point.

EXAMPLE: $$ \begin{align} &y(x)=x^2+3 \text{ is the unique solution to the initial value problem } \\ &\frac{dy}{dx}=2x \text{ and } y(1)=4 \end{align} $$ A function which solves a differential equation can be given in two different forms. An explicit solution is a function written in the usual form of a formula applied to an independent variable.

EXAMPLES: The following functions are defined explicitly $$ \begin{align} y(x)&=x^2+\tan(x) \\ y(x)&=\sqrt{1-x^2} \\ y(x)&=-\sqrt{1-x^2} \end{align} $$ A second way to write the function is to give a relationship between the function and the independent variable. The implicit function theorem can then be used to show a function exists which satisfies the relationship, though it may be difficult or impossible to write the function explicitly (the implicit function theorem is usually covered in Advanced Calculus 2).

EXAMPLES: $$ \begin{align} y-x^2-\tan(x)&=0 \\ x^2+y^2&=1 \end{align} $$ The first relation above generates the first function in the previous example while the second relation above generates both the second and the third functions of the previous example. An implicit solution to a differential equation is a relation which generates a function which solves the equation. Note that it is possible to check if an implicit solution solves a differential equation without actually computing the explicit solution by using implicit differentiation to solve for $y'$. While an explicit solution has obvious advantages over an implicit solution, often it is easy to find an implicit solution while the algebra involved in generating an explicit solution is difficult or impossible. It is also possible that an implicit solution will be more familiar and easier to understand ($x^2+y^2=1$ vs. $y=\pm\sqrt{1-x^2}$). In these cases we will settle for an implicit solution.

We will discuss more in the labs about the geometric interpretation of first order differential equations and how to use this interpretation to approximate solutions to equations we are unable to solve (which is most equations unfortunately).