# Preface

This section gives some motivating examples that lead to first order ordinary differential equations.

Differential equations are of central importance in modeling problems in engineering, the natural sciences and the social sciences. In physics, the knowledge of the force in an equation of motion usually leads to a differential equation. Thus, almost all the elementary and numerous advanced parts of theoretical physics are formulated in terms of differential equations. The virtue of differential equations models rests in their ability to capture the time-evolution of processes that exhibit elements of (instantaneous) feedback. Many of the principles, or laws, underlying the behavior of the natural world are statements or reactions involving rates at which things happen. When expressed in mathematical terms, the relations are equations containing derivatives. Historically, the first equations that were studied in 18th century for modeling mechanical problems were expressed through differentials---infinitesimal quantities. This is the reason why the equations containing derivatives are referred to as differential equations but not derivative equations. The subject of differential equations is one of the most interesting and useful areas of mathematics and its applications. Technology can prove very useful when studying differential equations.

Most fundamental laws of science are based on models that illustrate variations in physical properties and states of systems that include their rates of change. Here are several well-known examples

Description Model Application
Radioactive decay: $$\displaystyle \frac{{\text d}c}{{\text d}t} = - \lambda\,c(t)$$ substance amount c(t), λ is a parameter
Newton's second law: $$\displaystyle m\,\frac{{\text d}v}{{\text d}t} = F$$ particle of mass (m), having velocity (v), and applied force (F)
Malthusian growth model: $$\displaystyle \frac{{\text d}P}{{\text d}t} = r\, P(t)$$ population P(t) at time t and r is intrinsic rate
Fourier heat law: $$\displaystyle q = -\kappa\,\frac{{\text d}T}{{\text d}t}$$ heat flux q(t) at time t,T(t) is temperature, and κ is thermal conductivity
Fick's laws of diffusion: $$\displaystyle j = -D\,\frac{{\text d}c}{{\text d}t}$$ mass flux j(t) at time t, c(t) is concentration, and diffusion coefficient D
Inductance law: $$\displaystyle v(t) = L\,\frac{{\text d}i}{{\text d}t}$$ voltage in the circuit, v(t), inductance L, and current i(t)

Before embarking on a serious study of differential equations, you should gain some ideas why differential equations are so important and how you would benefit of its knowledge. We begin our journey of study with first order differential equations that involve a derivative of unknown function. This tutorial contains many practical problems for which differential equations are successfully utilized. A differential equation that describes some physical process is often called a mathematical model of the process, and many such models are discussed through this tutorial. It is noteworthy to discuss some elementary models that lead to first order differential equations.

Example: Radiocarbon dating is a method that provides objective age estimates for carbon-based materials that originated from living organisms. An age could be estimated by measuring the amount of carbon-14 present in the sample and comparing this against an internationally used reference standard. Cosmic ray protons blast nuclei in the upper atmosphere, producing neutrons which in turn bombard nitrogen, the major constituent of the atmosphere . This neutron bombardment produces the radioactive isotope carbon-14. The radioactive carbon-14 combines with oxygen to form carbon dioxide and is incorporated into the cycle of living things.

The method was developed in the late 1940s at the University of Chicago by the American physical chemist Willard Libby (1908--1980), who received the Nobel Prize in Chemistry for his work in 1960. It is based on the fact that radiocarbon (14C) is constantly being created in the atmosphere by the interaction of cosmic rays with atmospheric nitrogen. The resulting C14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis; animals then acquire C14 by eating the plants. When the animal or plant dies, it stops exchanging carbon with its environment, and thereafter the amount of C14 it contains begins to decrease as the C14 undergoes radioactive decay. Measuring the amount of C14 in a sample from a dead plant or animal, such as a piece of wood or a fragment of bone, provides information that can be used to calculate when the animal or plant died. The older a sample is, the less C14 there is to be detected, and because the half-life of C14 (the period of time after which half of a given sample will have decayed) is about 5,730 years by the emission of an electron of energy 0.016 MeV, the oldest dates that can be reliably measured by this process date to approximately 50,000 years ago, although special preparation methods occasionally permit accurate analysis of older samples.

The carbon-14 forms at a rate which appears to be constant, so that by measuring the radioactive emissions from once-living matter and comparing its activity with the equilibrium level of living things, a measurement of the time elapsed can be made. As soon as a living organism dies, it stops taking in new carbon. The ratio of carbon-12 to carbon-14 at the moment of death is the same as every other living thing (about 1.3 × 10-12), but the carbon-14 decays and is not replaced. The carbon-14 decays with its half-life of 5,700 years, while the amount of 6C12 remains constant in the sample. By looking at the ratio of carbon-12 to carbon-14 in the sample and comparing it to the ratio in a living organism, it is possible to determine the age of a formerly living thing fairly precisely. So let c(t) be the concentration of radioactive C14 in dead organic material at time t, counted since the time of death. Then c(t) obeys the following differential equation subject to the initial condition

$\frac{{\text d}c(t)}{{\text d}t} = -k\,c(t), \qquad c(0) = c_0 .$
Although you probably don't know how to solve this differential equation, you can rely on me or a software package and obtain
$c(t) = c_0 e^{-k\,t} .$
The value of k can be determined from the half-live of radioactive carbon-14:
$c_0 e^{-k\,5730} = \frac{1}{2}\, c_0 \qquad \Longrightarrow \qquad e^{-k\,5730} = \frac{1}{2} .$
Taking natural logarithm, we get its value
$k = -{\ln 0.5}{5730} \approx 0.000120968.$
Now we can determine the time of death:
$t = \frac{1}{k}\,\ln \frac{c_0}{c(t)} .$
For example, if you know that a fossil contains 25% of the original amount of C14, you use the equation above and obtain
$t = \frac{1}{0.000120968}\,\ln \frac{1}{0.25} \approx 11,460 \mbox{ years}.$
A nice feature of the method is that we do not need to know the absolute measurement of C14 but only a relative one.
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If a radioactive material, each atom has a certain probability of disintegrating into a lower state. For example, radium goes through a number of states the last of which is lead. If you have twice as much material, twice as many atoms will disintegrate, on the average, in a given time interval. If one has three times as much material, three times as many will disintegrate, and so on. These considerations lead to the differential equation
$\frac{{\text d}A}{{\text d}t} = - k\,A , \qquad k \mbox{ is a positive constant},$
where A(t) is the amount of material at time t. The constant k is positive because the amount A decreases with time---the differential equation describes decay not growth. A measure of the rate of decay is given by the half-life, t1/2. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive, radioactive decay. The term is also used more generally to characterize any type of exponential or non-exponential decay. Then A(t) can be expressed explicitly:
$A(t) = A(0) \,2^{-t/t_{1/2}} .$
So half-life is the time needed to reduce the amount A to A/2.    ■
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Example: Newton's law of cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. Sir Isaac Newton did not originally state his law in the above form in 1701 ("Scala graduum Caloris. Calorum Descriptiones & signa." in Philosophical Transactions, volume 22, issue 270), when it was originally formulated. Rather, using today's terms, Newton noted after some mathematical manipulation that the rate of temperature change of a body is proportional to the difference in temperatures between the body and its surroundings.

Sometime when we need only approximate values from Newton’s law, we can assume a constant rate of cooling, which is equal to the rate of cooling corresponding to the average temperature of the body during the interval. Let u(t) be the temperature of the object at time t; for example, you have a mug of coffee, assuming that the its temperature remains uniform. Then Newton's law of cooling is read

$\frac{{\text d}u}{{\text d} t} = k \left( T - u(t) \right) ,$
where k is a constant cooling coefficient, and T is the temperature of surroundings, which must remain constant during the cooling of the coffee. If the temperature of coffee is higher than the surrounding temperature T (which is assumed), then the coefficient k is positive. You will learn shortly in section on linear equations that its solution is
$u(t) = T + \left( u(0) - T \right) e^{-kt} \quad \to \quad T \qquad \mbox{as} \quad t\to \infty .$
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