What precisely is a differential equation?

Definition of sociology by different authors

An equation is a mathematical sentence, of sorts, that describes the relationship between two or more things.

If you say a = b, then youve just described the (trivially simple) relationship between a and b.

A differential equation is a particular type of equation that includes not just things, but the rates of change of things.

,For example, consider the simple fact that if you drive at 50 miles an hour for one hour, you will have driven 50 miles.

Thats so obvious as to seem intuitive, because you already have a deep working understanding of how cars and their speedometers work.

,But if we look at the equation a little more deeply, we can see some interesting things.

Your position (50 miles from some chosen starting point) is a function of both time (how long you drive) and the rate of change of your position with respect to time.

Were here describing a relationship between a thing and the rate of change of that thing: position and the rate of change of position with respect to time.

,Why is this an interesting insight? Because what if you didnt drive at a constant speed? What if your speed varied along the trip? Then figuring out how far you went isnt so simple as multiplying a number u2014 your constant speed u2014 times the time you spent driving.

Its a harder problem to solve.

(And in fact, solving the problem involves understanding the relationship between your rate-of-change-of-speed and time, turning it into an equation, and then integrating that equation over the time you were driving u2026 but thats just a sidebar, and not directly relevant to your question.

),These kinds of equations u2014 where a thing is related to its own rate of change u2014 come up all the time both in nature and in what Ill call the invented sciences, things like sociology and economics.

,For example, you have a plate of hot soup sitting on the table.

How long will it be before the soup is room temperature? One might naively guess that the soup will lose heat to the environment at a constant rate until it reaches room temp, but this turns out to be wrong.

It turns out the rate at which the soups temperature changes depends on the soups temperature.

So you need to use a differential equation to predict what the temperature will be at some given time, or how long itll take for the temperature to reach some chosen value.

,A different, but fundamentally similar, example, is this: Say you have some bacteria in a petri dish, and those bacteria multiply periodically.

How many bacteria will you have tomorrow? That depends on the rate of change of the population over time u2026 and in fact, in that case it turns out the rate of change of the population is itself related to the population.

(Expand on those ideas u2014 where the rate of change of a thing depends on the thing itself u2014 for a little bit and you stumble across the exponential function e to the x, which is so rich and widely applicable as to basically be its own sub-branch of mathematics.

),But getting back to basics, a differential equation is any equation that includes both things and also rates of change of those things.

Theyre worth studying in the abstract because you cant (generally) use algebra to solve equations like that; you need to use different methods, and those methods are worth learning a bit about.

Importance of sociology

Generally speaking, it attempts to explain the outcomes of individuals and their personal attributes, opinions, tastes, and characteristics in terms of their social environment.

More specifically, it is an application of statistical analysis and the search for cause and effect relationships between identifiable social variables like race, religion, income, earnings, education, gender, etc.

,The field is broad, however, and a lot of sociological literature can easily delve into subject matter that is classifiable as psychology, history, philosophy, economics, and theories of knowledge.

,Ultimately, what Sociology is for is to help explain the human condition by making reference to our self-created environments and understanding how the reflexive nature of social action contributes to the formation of personal identity, intersubjective meanings, beliefs and cultures, institutional arrangements and political structures.

Nature of sociology

The nature of Sociology is as follows:,(1) Sociology is an independent science:Sociology is not treated and studied as a branch of any other science like philosophy, history.

Now it has emerged into an independent science.

As an independent science it has its own field of study.

,(2) Sociology is a social science and not a physical science:All the sciences are divided into two categories: natural sciences and social sciences.

Natural sciences study physical phenomena where as social sciences study social phenomena.

Social sciences include Economics, Political Science, and Anthropology etc.

Sociology belongs to the family of social sciences.

As a social science it concentrates its attention on man, his social behaviour, activities and social life.

In other words, it studies man as a social being.

,(3) Sociology is a pure science and not an applied science:The aim of applied science is to apply the acquired knowledge into life and to put it to use.

But the aim of pure sciences is the acquisition of knowledge and it is not bothered whether the acquired knowledge is useful or can be put to use.

Sociology is a pure science, because it aims at the acquisition of knowledge about human society, hot the utilisation of the knowledge.

,(4) Sociology is an abstract science and not a concrete science:This doesnt mean that Sociology, is an art and not a science.

It only refers that Sociology is not interested in concrete manifestations of human events.

It is more concerned with the form of human events and their patterns.

Similarly, Sociology does not confine itself to the study of this society or that particular society.

It simply means that Sociology is an abstract science, not a concrete science.

,(5) Sociology is a categorical and not a normative discipline:Sociology confines itself about what is, not what should be or ought to be.

As a science it is silent about questions of value.

It does not make any kind of value judgment.

It only means Sociology as a discipline cannot deal with problems of good and evil, right and wrong.

,(6) Sociology is a generalising and not a particularising science:Sociology does not study each and every event that takes place in society.

It makes generalization on the basis of some selected events.

For example, not by studying or examining all the secondary groups but by observing a few secondary groups, a sociologist makes generalization of secondary groups.

,(7) Sociology is a general science and not a special social science:The area of inquiry of Sociology is general and not specialised.

Social sciences like Political Science, History, Economics, etc.

study human interaction but not all about human interactions.

But Sociology does not investigate special kind of phenomena in relation to human life, and activities but it only studies human activities in a general way.

,(8) Sociology is both a rational and empirical science:Empiricism is the approach that emphasizes experiences and the facts that result from observation and experimentation.

On the other hand, rationalism stresses reason and theories that result from logical inference.

The empiricist collects facts, the rationalist co-ordinates and arranges them.

In sociological theory both are significant.

Thus, Sociology is both a rational and empirical science.

,ConclusionThus, from the above discussion we come to know that the nature of Sociology is independent, social, a categorical, pure, abstract, and generalizing; both are a rational and an empirical social science.