What is a Differential Equation?

Section 2.1 What is a Differential Equation?

Subsection 2.1.1 Equations and Questions

Definition 2.1.1.

A differential equation (DE, for short) is any equation involving a function \(f(t)\) and its derivatives \(f^\prime(t)\text{,}\) \(f^{\prime \prime}(t)\text{,}\) etc. The highest derivative in the equation is called the order of the equation. The function \(f\) can have one variable as written here; in this case the equation will be called an ordinary differential equation (ODE for short). If the function \(f\) depends on more than one variable, then all the various partial derivatives can also be part of the DE. In this case, the equation is called a partial differential equation (PDE for short).

More holistically, an equation is always implicitly a question: what thing satisfies a given relationship? In algebra, equations ask about numbers. \(t + 5 = 7\) implicitly asks: which number, when replacing \(t\text{,}\) satisfies the equation? The answer, of course, is \(t=2\text{;}\) the solution to the equation gives the correct replacement.

For differential equations, I still have an equation, but the question is different. Instead of asking for a number, I want to know what function satisfies the equation. If the equation is \(f^\prime(t) = 3 f(t)\text{,}\) then the equation is asking this question; what function has a derivative which is three times the original function?

Subsection 2.1.2 Notations and Definitions

There are many ways to write a differential equation. The theory allows for any strange and complicated expression in a function and its derivatives.

\begin{equation*} \frac{f(t) f^\prime(t) + e^t \cos t}{f^{\prime \prime}(t)} - \ln \left( \frac{1}{1 + (f(t))^6} \right) - \sqrt{f^\prime(t)} = 17 \end{equation*}

This expression is a differential equation, but it is basically nonsense. I have no idea if it has a solution, but I am certain that this equation doesn’t relate to any concrete motivating problem. Most of the DEs we deal with are those that arise in applied mathematics, that deal with some problem that we are actually interested in. These, for the most part, have reasonable forms. Let me define some of these forms for you.

Definition 2.1.2.

Let \(t\) be an independent variable and \(y(t)\) a function. A differential equation is called autonomous is the independent variable doesn’t appear. For a first order equation, this looks like

\begin{equation*} \frac{dy}{dt} = f(y) \end{equation*}

for some expression \(f(y)\) in the dependent variable. Note that \(y(t)\) is the function I am looking for; \(f(y)\) is also a function, but I’m only using that function to say that the right side of the equation is some expression in the varialbe \(y\text{.}\)

Definition 2.1.3.

An first order equation is called linear if it has this form, where \(y(t)\) is the potential solution but \(a(t)\text{,}\) \(b(t)\) and \(c(t)\) are any other function in the same independent variable.

\begin{equation*} a(t) \frac{dy}{dt} + b(t) y(t) = c(t) \text{.} \end{equation*}

Likewise, a second equation is called linear if it has this form.

\begin{equation*} a(t) \frac{d^2y}{dt^2} + b(t) \frac{dy}{dt} + c(t) y(t) = d(t) \end{equation*}

The order of a linear equation can be as high as you want. This means the full general form of a linear equation looks sort-of like a polynomial, with ‘powers’ as order of differentiation and coefficients as other functions.

\begin{equation*} a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \ldots + a_1(t) y^\prime + a_0(t) y = f(t)\text{.} \end{equation*}

Finally, a linear equation is called homogenous if the term which is not multiplied by any derivative of the function (\(c(t)\) in the first order equation, \(d(t)\) in the second order equation) \(0\text{.}\)

A very useful notational tool for linear DEs is the idea of a linear differential operator. In calculus, I used this language to talk about \(\frac{d}{dt}\) as the the thing---the agent---that will take the derivative (in \(t\)) of any function it comes across. This language of differential operators is even more useful in this course.

Definition 2.1.4.

A differentiable operator is an operator which acts on functions either by multplication by other functions or by differentiation. The operator is linear if the various pieces of the operator are added together; equivalently, if the operator respects addition and scalar multiplication.

Consider the second order linear equation that was written in the previous definition. The function \(y(t)\) and its derivatives only show up on the left side of the equation.

\begin{equation*} a(t) \frac{d^2y}{dt^2} + b(t) \frac{dy}{dt} + c(t) y(t) = d(t) \end{equation*}

This means I can think of the entire left side as something that happens to the function \(y(t)\text{.}\) I can express this with a linear differential operator.

\begin{equation*} L = a(t) \frac{d^2}{dt^2} + b(t) \frac{d}{dt} + c(t) \end{equation*}

Using this operator, I can write the differential equation very succinctly. This second order equation now becomes \(Ly = d(t)\text{.}\) I’ll make frequent use of this kind of operator notation throughout the course.

Subsection 2.1.3 The Most Important Example

Figure 2.1.5. The family \(f(t) = ce^{\alpha t}\) where \(c \in \RR\text{.}\)

The simplest differential equation is

\begin{equation*} \frac{df}{dt} = \alpha f(t)\text{.} \end{equation*}

For many differential equations, I seek a translation of their meaning. The translation of this simple equation is that the rate of change is propotional to the current value. This is a statement of percentage growth; the proportionality constant is the percentage.

The solutions to this equation are are \(f(t) = ce^{\alpha t}\text{.}\) Notice that there is a constant, \(c\text{,}\) such that there will be a whole infinite family of solutions. A specific solution is specified by a choice of \(c\text{.}\)

Percentage growth applies to population models. If a population is described by the solution \(f(t) = ce^{\alpha t}\) then \(f(0) = ce^0 = c\) and I must interpret the constant \(c\) as the population when \(t=0\text{.}\) This is the starting popluation, so I call \(c\) the initial value of the solution. A full solution of a differential equation will usually consist of a function and choice(s) for the initial value(s).

Definition 2.1.6.

A differential equation along with a specified initial value is called an initial value problem or IVP.

If I don’t make a choice, as I said, I get an infinite family of solutions. I can visualize this family as a series of graphs in \(\RR^2\text{.}\) Figure 2.1.5 shows the graphs for \(f(t) = ce^{\alpha t}\text{.}\)

Subsection 2.1.4 Archtypical Examples

I want to understand the behaviour of solutions of DEs. The examples in this section give some typical behaviours and archetypes, ideas that will be repeated over and over again in the course. For these examples, I’m simply going to give you the DE and its solution — no solution methods here. The point is to show the kinds of behaviours that often occur.

Example 2.1.7.

\begin{equation*} \frac{d^2f}{dt^2} + 9 f = 0 \end{equation*}

This is solved by \(\sin 3t\) and also by \(\cos 3t\text{.}\) Moreover, any linear combination \(a\sin 3t + b \cos 3t\) is also solution. In a second order equation, I expect two linearly independent solutions and the general solution is a linear combination of the two linearly independent solutions. (This kind of description of solution is where the linear algebra notions of linear combinations and linear indenpendence are most useful.)

Example 2.1.8.

\begin{equation*} \frac{dy}{dt} = t \sqrt{y} \end{equation*}

This is solved by \(y = (\frac{t^2}{4} + c)^2\text{,}\) which is a nice family with one real parameter. However, this is also solved by \(y=0\text{,}\) even though it isn’t in the family (there is no value of \(c\) that will lead to the solution \(y=0\)). It’s important to be aware that this situation can occur. There is even a name for such a solution.

Definition 2.1.9.

An extraneous solutions to a DE, one which falls outside families with parameters, is called singular solution.

Example 2.1.10.

\begin{equation*} t \frac{dy}{dt} = 4y \end{equation*}

This is solved by \(y = ct^4\text{,}\) which is another reasonable family. However, there is a strange, singular solution.

\begin{equation*} y(t) = \left\{ \begin{matrix} -t^4 \amp t \leq 0 \\ t^4 \amp t>0 \end{matrix} \right. \end{equation*}

The derivatives of this function line up at the origin, so the function is of class \(C^1\) (actually \(C^3\text{,}\) if you work it out).

Example 2.1.11.

\begin{equation*} \frac{dy}{dx} = \frac{-x}{y} \end{equation*}

The curve \(x^2 + y^2 = c\) solves this equation implicitly. I could break this up into two functions, but its much more natural to leave it as an implicit locus, in this case, a circle. This is very typical: often solutions are left in an implicit form as loci, even though in theory I always look for solutions which are functions \(y = f(x)\text{.}\) Also notice in this example that only non-negative \(c\) values are allowed in this family of solutions. There is no guarantee that all values of a parameter will lead to reasonable solutions.

Subsection 2.1.5 Pure and Applied Perspectives

I will be looking at differential equations from both pure mathematics and applied mathematics points of view. The pure mathematician is interested in the following kinds of questions.

When does a solution exists?
Can I prove that a solution exists?
Is the solution unique?
Is there a complete family? How many parameters exist, and what are their domains?
Can I write and prove theorems to answer these questions?

In contrast, the applied mathematician is interested in the following kinds of questions.

How many solutions fit the data or initial values?
How do the solutions grow? What is their behaviour?
Are the solutions stable?
How difficult are the solutions to calculate? Can I calculate them exacly, or only approximately?
Can I answer questions about the model even without an explicit solution?

Thinking as an applied mathematician, a DE is an mathematical version of a scientific method. Often an observation about a phenomenon can be expressed as a relationship between a function and its derivative, such as the observation of percentage growth. The DE, then, is the hypothesis born of observation. If I can find the solution, it gives me a predictive model of the phenomenon, which I can test. If the solution matches the observed behaviour, I conclude the DE model is relatively reliable; if the solution diverges from the observed behaviour, I discard or amend the DE.

In this way, DEs allow the modelling many phenomena: popluations, radioactive decay, cooling, disease, metabolish, newtonian motion with friction, chemical reactions, gravity, predaor-prey models, hamiltonian mechanics, quantum mechanics, interest, bacterial growth, neuron firing, ecology, mixtures, draining, series circuits, suspended cables, and many, many more.

A robust study of differential equations involves both the pure and applied perspectives. The pure perspective gives background, context and rigour. The theorems it produces are necessary for the efficient use of DEs in applied mathematics. The applied perspective, most often, gives the motivation and selects the important problems. It also stresses the value of qualitative methods and their interpretation.

Subsection 2.1.6 The Scarcity of Solutions

Even going briefly into the study of DEs leads to one immediate conclusion: DEs are terribly difficult to solve. The sad truth is that mathematicians can produce exact solutions to DEs only a very small portion of them. Due to this limitation, many techniques are developed to understand approximate solutions or infer information about solutions indirectly.

Alternatively, if mathematicians are unable to solve a differential equation directly because there isn’t a function in our arsenal that solves it, there is another radical option: invent a new function. In this way, DEs can be the source of many new types of functions. It’s not uncommonly that even a very reasonable DEs which itslef only involves elementary functions can have a solution which is a completely new, unknown kind of function.

Example 2.1.12.

\begin{equation*} \frac{dy}{dt} = e^{-t^2} \implies y = \int e^{-t^2}dt + c \end{equation*}

This integrand is a \(C^\infty\) function, so the anti-derivative exists, but there is no name for it in the elementary functions. Therefore, the solution is some new function which, if I wanted, I could name and then try to investigate.

This example leads me to an important question: what do I actually mean by ‘solving a differential equation’? The naivë understanding of solving answers something like this: find a function among the functions I know that fits the equation. Indeed, this is how I talked about solving in Subsection 2.1.1. In this sense, most differential equations are ‘unsolvable’. However, after the discussion of this section, I can give a new understanding of ‘solving’. To solve a differential equation is to describe the function that fits the equation. In this sense, the situation is much more optimistic: almost any differential equation I can write down will have a solution, in this sense. However, the difficulty is now in describing the function. There is no guarantee that the function is one that I already know. It might be something completely new. How do I investigate its properties, graph it, understand its behaviour?

Finally, this is an important example of the kinds of linguistic subtleties that run thought mathematics. Even the very basic term ‘solve’ can mean different things depending on our assumptions. To deeply understand mathematics is to be aware of all of the underlying assumption that we bring to the discipline.

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