First-Order Differential Equations | Section 1: An Introduction

Mathematics is the language of the universe. From physics and chemistry to business and sociology, it lets us understand and describe the world around us. So being able to speak it and read it, adds a new flavor and color to this world. While static problems are definitely important to understand, I think that the more interesting ones have to do with the natural phenomena that involves change. And that is exactly where the field of Differential Equations comes in. It describes the ever changing universe that we live in.

This series of posts is going to be the first series I create on the topic of differential equations. Ordinary Differential Equations (ODE) will be the main focus of this series. And will dive deep into the mathematical proofs and theorems behind what you learn in your first university class on differential equations.

This series   assumes that you already know the basics of calculus (i.e derivatives, integrals). Later on we will also be needing some basic linear algebra . But do not worry if you are not very familiar with both of these topics as I will be doing my best to explain things as we go along.

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First-Order Differential Equations: What are they all about?

A big part of this series will focus on First-Order ODE and the Second-Order ODE. Therefore I think that it would be more appropriate if we began by defining those two terms first.

The first-order ODE consist of an unknown function x(t) (here we took a function x in terms of the time variable t. Of course you can name the function whatever you like) and its first derivative x’(t)=dx/dt as shown in the example equation

Finding the derivative of a function is the same as finding the slope of the function. And of course, as you may as well know that the slope of the function in physics is called the velocity.

The second-order ODE consist of an unknown function y(x) (note that this time we named our function y in terms of the variable x). Its first derivative y’(x)=dy/dx and its second derivative y’’(x)=d²y/dx² as shown in the example equation

Finding the second derivative tells you about the bending of the function. Additionally, finding the bending of the function in physics is also finding the acceleration.

What we will be doing by studying differential equations can be summed up with three points:

• Take a specific physical phenomena or situation and find the differential equation that best describes it,
• then either compute or solve that equation — either precisely or approximately.
• And finally interpret the solution obtained and draw a conclusion.

The field of algebra seeks to find the unknown numbers that satisfy equations such as x³+7x²-11x +41=0. Whereas by solving differential equations, we are tasked with finding all the unknown functions (if possible) for which its identity holds on some interval of real numbers. Let us take a simple example to solidify this idea and make it more concrete.

Example 1

Consider the following function

where C is a constant. Then the first-order ODE of (1) is

Thus every function of the form (1) is a solution of the ODE which we have solved above

In other words, (1) defines an infinite family of different solutions to (2). With a solution being for every unique choice of the arbitrary constant C.

Example 2

Now let us take an example derived from the real world: consider that the time rate of change of a population P(t) — with constant birth and death rates, in our simple case — is proportional to the size of the population P. In other words, the following first-order ODE reflects our simple example

where k is the constant of proportionality. Now we note that each function of the form

is a solution to (3). From which we can verify

that it in fact is a solution of (3). Now, even if the value of the constant k is known in (4), the ODE (4) has infinitely many solutions of the form (3) — one for each unique choice of the constant C.

This is actually typical for differential equations, it allows us to use additional information to pick one of the solutions and use that obtained solution to solve for the given situation: For instance consider a population of a colony of bacteria at time t. And suppose that at t=0 the population P(t) was 1000. Now assume that the population has doubled after 1 unit of time. Our goal here is to solve (4) for the extra information given. Thus, this new information about P(t) gives us

therefore it follows that

with the obtained values of C and k, we can replace them in (3) and (4) to get

And now, with equation (5) we can predict the population of our colony of bacteria in any point in time, by just substituting t with the desired value. For instance, at time t=2.5, the population of the colony of bacteria is P(2.5)= 5656.

Figure 1 show various different graphs of P(t) with k=ln2. If we were to plot all the graphs of dP/dt=kP, they will in fact fill out the entirety of the 2-dimensional plane, and not one graph will overlap the other. Meaning no two graphs will intersect. Moreover, selecting a point in the plane P₀ amounts to a determination of P(0), since any given point on the plane passes through exactly one solution.

It is helpful to note that P(0)=1000 in example 2 is called the initial condition, and t=0 is the starting time.

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Mathematical Modeling

Our discussion of population growth in example 2 illustrates the importance of mathematical modeling. A mathematical model can be broken down into three parts:

1. Problem formulating in the mathematical language; In other words, constructing the mathematical model.
2. Analyzing the results of the mathematical model that has been created.
3. And finally, interpreting the results gotten from our analysis and applying them to the initial real world situation; In other words we are trying to answer the questions which were originally posed.

Going back to example 2 and relating it to the mathematical model which I’ve just explained: Our real world problem is to be able to predict the population number in a given time in the future. Our mathematical model which we came up with consists of variables (P and t) that describe the given problem, along with one or more questions relating these variables (dP/dt=kP, P(0)=P₀) that are known or are assumed to hold. Then, the mathematical analysis consists of solving these questions (in our case, solving for P as a function of t). Then finally, we apply these results and attempt to solve the original question.

As a further explanation of the process, let’s go back to example 2: Think of formulating the mathematical model consisting of the equations dP/dt=kP, P(0)=1000 for describing the bacteria population. Then the mathematical analysis consists of solving for the solution function (5) as our mathematical result. And finally, for an interpretation in terms of the real world situation, we substituted t=2.5 to obtain the predicted population of P(2.5)=5656 after 2.5 units of time. If the colony of bacteria is growing under an unlimited source of food and space (which is the ideal condition), then our prediction may be accurate. In which case we conclude that our mathematical model is sufficient for studying this particular population.

And finally, for an interpretation in terms of the real world situation, we substituted t=2.5 to obtain the predicted population of P(2.5)=5656 after 2.5 units of time. If the colony of bacteria is growing under an unlimited source of food and space (which is the ideal condition), then our prediction may be accurate. In which case we conclude that our mathematical model is sufficient for studying this particular population.

On the other hand, it may turn out that no solution of the selected differential equation accurately fits the actual population we’re studying. For instance, for no choice of C and k does the solution in equation (4) accurately describe the actual growth of the human population of the world in the past few centuries. Therefore we must conclude that the differential equation (3) is insufficient for modeling the world population.

With sufficient insight, we might formulate a new mathematical model including perhaps a more complicated differential equation, one that takes into account such factors as a limited food supply and the effect of increased population on birth and death rates. With the formulation of this new mathematical model, we may attempt to traverse once again the diagram of figure 2. If we can solve the new differential equation, we get new solution functions to compare with the real world population.

Indeed, a successful population analysis may require refining the mathematical model still further as it is repeatedly measured against real-world experience which will require traversing the diagram again and again until we have gotten a model that is adequate enough and accurately predicts the world population.

If we can solve the new differential equation, we get new solution functions to compare with the real world population. Indeed, a successful population analysis may require refining the mathematical model still further as it is repeatedly measured against real-world experience which will require traversing the diagram again and again until we have gotten a model that is adequate enough and accurately predicts the world population.

But in example 2 we simply ignored any complicating factors that might affect our bacteria population. This made the mathematical analysis quite simple, perhaps unrealistically so. A satisfactory mathematical model is subject to two contradictory requirements: It must be sufficiently detailed to represent the real world situation with relative accuracy, yet it must be sufficiently simple to make the mathematical analysis practical.

If the model is so detailed that it fully represents the physical situation, then the mathematical analysis may be too difficult to carry out. But if the model is too simple, the results may be so inaccurate as to be useless. Thus there is an inevitable tradeoff between what is physically realistic and what is mathematically possible. The construction of a model that adequately bridges this gap between realism and feasibility is therefore the most crucial and delicate step in the process. Ways must be found to simplify the model mathematically without sacrificing essential features of the real world situation.

The construction of a model that adequately bridges this gap between realism and feasibility is therefore the most crucial and delicate step in the process. Ways must be found to simplify the model mathematically without sacrificing essential features of the real world situation.

. . .

In the next section, we will start diving into the topic of differential equations and start looking at ways — in particular, integrals as general and particular solutions — to solving them while also applying those ways to real world problems as we go along.

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