Preface
This section gives some motivating examples that lead to first order ordinary differential equations.
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Glossary
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.
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 (^{14}C) is constantly being created in the atmosphere by the interaction of cosmic rays with atmospheric nitrogen. The resulting C^{14} combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis; animals then acquire C^{14} by eating the plants. When the animal or plant dies, it stops exchanging carbon with its environment, and thereafter the amount of C^{14} it contains begins to decrease as the C^{14} undergoes radioactive decay. Measuring the amount of C^{14} 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 C^{14} there is to be detected, and because the half-life of C^{14} (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 _{6}C^{12} 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 C^{14} 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
============== to be worked out ==============
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
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
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