I think we should start with the history of such a glorious mathematical tool as differential equations. Like all differential and integral calculus, these equations were invented by Newton at the end of the 17th century. He considered this particular discovery so important that he even encrypted a message that today can be translated approximately like this: "All laws of nature are described by differential equations." This may seem like an exaggeration, but it is. Any law of physics, chemistry, biology can be described by these equations.
A huge contribution to the development and creation of the theory of differential equations was made by mathematicians Euler and Lagrange. Already in the 18th century, they discovered and developed what they are now studying in senior university courses.
A new milestone in the study of differential equations began thanks to Henri Poincaré. He created a “qualitative theory of differential equations”, which, in combination with the theory of functions of a complex variable, made a significant contribution to the foundation of topology - the science of space and its properties.
What are differential equations?
Many are afraid of the single phrase "differential equation." However, in this article we will describe in detail the whole essence of this very useful mathematical apparatus, which is actually not as complicated as the name suggests. In order to start talking about first-order differential equations, one should first become familiar with the basic concepts that are inherently connected with this definition. And we will start with the differential.
Differential
Many have known this concept since school. However, we will dwell on it in more detail. Imagine a graph of a function. We can increase it to such an extent that any segment of it takes the form of a straight line. On it we take two points that are infinitely close to each other. The difference in their coordinates (x or y) will be infinitely small. It is called the differential and is denoted by the signs dy (differential from y) and dx (differential from x). It is very important to understand that the differential is not a finite quantity, and this is its meaning and main function.
And now it is necessary to consider the following element, which is useful to us in explaining the concept of a differential equation. This is a derivative.
Derivative
We all probably heard in school and this concept. The derivative is said to be the rate of growth or decrease of a function. However, much of this definition becomes incomprehensible. Let's try to explain the derivative through differentials. Let's go back to an infinitely small segment of a function with two points that are at a minimum distance from each other. But even for this distance, the function manages to change by some amount. And in order to describe this change, they also came up with a derivative that could otherwise be written as the ratio of differentials: f (x) '= df / dx.
Now it is worth considering the basic properties of the derivative. There are only three of them:
- The derivative of the sum or difference can be represented as the sum or difference of the derivatives: (a + b) '= a' + b 'and (ab)' = a'-b '.
- The second property is related to multiplication. The derivative of a product is the sum of the products of one function and the derivative of another: (a * b) '= a' * b + a * b '.
- The derivative of the difference can be written in the form of the following equality: (a / b) '= (a' * ba * b ') / b 2 .
All these properties are useful to us for finding solutions of first-order differential equations.
There are also private derivatives. Suppose we have a function z that depends on the variables x and y. To calculate the partial derivative of this function, say, with respect to x, we need to take the variable y for a constant and simply differentiate it.
Integral
Another important concept is the integral. In fact, this is the exact opposite of the derivative. There are several types of integrals, but to solve the simplest differential equations we need the most trivial indefinite integrals.
So what is an integral? Suppose we have some dependence of f on x. We take the integral from it and get the function F (x) (often called the antiderivative), the derivative of which is equal to the original function. Thus F (x) '= f (x). It also follows that the integral of the derivative is equal to the original function.
When solving differential equations, it is very important to understand the meaning and function of the integral, since you will have to take them very often to find a solution.
The equations are different depending on their nature. In the next section, we will consider the types of differential equations of the first order, and then we will learn how to solve them.
Classes of Differential Equations
"Diffurs" are divided according to the order of the derivatives involved in them. Thus there is a first, second, third or more order. They can also be divided into several classes: ordinary and in private derivatives.
In this article, we consider ordinary first-order differential equations. Examples and methods for solving them will also be discussed in the following sections. We will consider only ODEs, because these are the most common types of equations. Ordinary are divided into subspecies: with separable variables, homogeneous and heterogeneous. Next, you will learn how they differ from each other, and learn how to solve them.
In addition, these equations can be combined so that after we get a system of differential equations of the first order. We will also consider such systems and learn how to solve.
Why do we consider only the first order? Because you need to start with a simple one, and to describe everything related to differential equations in one article is simply impossible.
Separated Variable Equations
These are perhaps the simplest differential equations of the first order. These include examples that can be written like this: y '= f (x) * f (y). To solve this equation, we need a formula for representing the derivative as the ratio of differentials: y '= dy / dx. Using it, we obtain the following equation: dy / dx = f (x) * f (y). Now we can turn to the method of solving standard examples: we divide the variables in parts, that is, we transfer everything from the variable y to the part where dy is located, and also do with the variable x. We get an equation of the form: dy / f (y) = f (x) dx, which is solved by taking the integrals from both sides. Do not forget about the constant that must be set after taking the integral.
The solution to any "diffusion" is a function of the dependence of x on y (in our case) or, if there is a numerical condition, then the answer is in the form of a number. Let us analyze the entire course of the solution using a concrete example:
y '= 2y * sin (x)
We transfer the variables in different directions:
dy / y = 2 * sin (x) dx
Now we take the integrals. All of them can be found in a special table of integrals. And we get:
ln (y) = -2 * cos (x) + C
If required, we can express "igrek" as a function of "x". Now we can say that our differential equation is solved if the condition is not specified. A condition can be specified, for example, y (n / 2) = e. Then we simply substitute the value of these variables in the solution and find the value constant. In our example, it is 1.
Homogeneous first order differential equations
Now move on to the more difficult part. First-order homogeneous differential equations can be written in general form as follows: y '= z (x, y). It should be noted that the right function of two variables is homogeneous, and it cannot be divided into two dependences: z on x and z on y. Checking whether the equation is homogeneous or not is simple enough: we make the substitution x = k * x and y = k * y. Now cut all k. If all these letters are abbreviated, then the equation is homogeneous and you can safely proceed to its solution. Looking ahead, let's say: the principle of solving these examples is also very simple.
We need to make a replacement: y = t (x) * x, where t is a certain function that also depends on x. Then we can express the derivative: y '= t' (x) * x + t. Substituting all this into our original equation and simplifying it, we get an example with separable variables t and x. We solve it and obtain the dependence t (x). When we received it, we simply substitute y = t (x) * x in our previous replacement. Then we obtain the dependence of y on x.
To make it clearer, consider an example: x * y '= yx * e y / x .
When checking with replacement, everything is reduced. So the equation is really homogeneous. Now we make another replacement, which we talked about: y = t (x) * x and y '= t' (x) * x + t (x). After simplification, we obtain the following equation: t '(x) * x = -e t . We solve the resulting example with separated variables and get: e -t = ln (C * x). We only need to replace t with y / x (because if y = t * x, then t = y / x), and we get the answer: e -y / x = ln (x * ).
First order linear differential equations
It's time to consider another broad topic. We will analyze inhomogeneous first-order differential equations. How do they differ from the previous two? Let's figure it out. First-order linear differential equations in general form can be written as follows: y '+ g (x) * y = z (x). It is worth clarifying that z (x) and g (x) can be constant values.
And now an example: y '- y * x = x 2 .
There are two ways to solve, and we will deal with both in order. The first is the method of varying arbitrary constants.
In order to solve the equation in this way, you must first equate the right side to zero and solve the resulting equation, which after the transfer of parts takes the form:
y '= y * x;
dy / dx = y * x;
dy / y = xdx;
ln | y | = x 2/2 + C;
y = e x2 / 2 * y C = C 1 * e x2 / 2 .
Now we need to replace the constant C 1 with the function v (x), which we have to find.
y = v * e x2 / 2 .
We replace the derivative:
y '= v' * e x2 / 2 -x * v * e x2 / 2 .
And substitute these expressions in the original equation:
v '* e x2 / 2 - x * v * e x2 / 2 + x * v * e x2 / 2 = x 2 .
You can see that on the left side two terms are abbreviated. If in some example this did not happen, then you did something wrong. Continue:
v '* e x2 / 2 = x 2 .
Now we solve the usual equation in which we need to separate the variables:
dv / dx = x 2 / e x2 / 2 ;
dv = x 2 * e - x2 / 2 dx.
To extract the integral, we have to apply integration by parts here. However, this is not the topic of our article. If you are interested, you can independently learn to perform such actions. It is not difficult, and with sufficient skill and attentiveness does not take much time.
Let us turn to the second method for solving inhomogeneous equations: the Bernoulli method. Which approach is faster and easier is up to you.
So, when solving the equation with this method, we need to make a replacement: y = k * n. Here k and n are some x-dependent functions. Then the derivative will look like this: y '= k' * n + k * n '. We substitute both substitutions into the equation:
k '* n + k * n' + x * k * n = x 2 .
Group:
k '* n + k * (n' + x * n) = x 2 .
Now we need to set to zero what is in parentheses. Now, if we combine the two resulting equations, we get a system of differential equations of the first order, which must be solved:
n '+ x * n = 0;
k '* n = x 2 .
The first equality is solved as an ordinary equation. To do this, separate the variables:
dn / dx = x * v;
dn / n = xdx.
Take the integral and get: ln (n) = x 2/2. Then, if we express n:
n = e x2 / 2 .
Now we substitute the resulting equality in the second equation of the system:
k '* e x2 / 2 = x 2 .
And transforming, we get the same equality as in the first method:
dk = x 2 / e x2 / 2 .
We will also not analyze further actions. It is worth saying that at first the solution of first-order differential equations causes significant difficulties. However, with a deeper immersion in the topic, it starts to turn out better and better.
Where are differential equations used?
Differential equations are very actively used in physics, since almost all the basic laws are written in differential form, and the formulas that we see are the solutions to these equations. In chemistry, they are used for the same reason: the basic laws are derived with their help. In biology, differential equations are used to model the behavior of systems, for example, a predator - a prey. They can also be used to create breeding models of, say, a colony of microorganisms.
How differential equations will help in life?
The answer to this question is simple: no way. If you are not a scientist or engineer, then they are unlikely to be useful to you. However, for general development it will not hurt to know what a differential equation is and how it is solved. And then the question of the son or daughter "what is a differential equation?" will not baffle you. Well, if you are a scientist or engineer, then you yourself understand the importance of this topic in any science. But the most important thing is that now to the question "how to solve a first-order differential equation?" You can always give an answer. Agree, it’s always nice when you understand something that people are even afraid to figure out.
The main problems in studying
The main problem in understanding this topic is a poor ability to integrate and differentiate functions. If you take derivatives and integrals poorly, then you probably should still learn, master the different methods of integration and differentiation, and only then proceed with the study of the material that was described in the article.
Some people are surprised when they learn that dx can be transferred, because earlier (at school) it was claimed that the fraction dy / dx is indivisible. Here you need to read the literature on the derivative and understand that it is a ratio of infinitesimal quantities that can be manipulated in solving equations.
Many do not immediately realize that the solution of first-order differential equations is often a function or an unshifted integral, and this error gives them a lot of trouble.
What else can be studied for a better understanding?
It is best to start further immersion in the world of differential calculus with specialized textbooks, for example, on mathematical analysis for students of non-mathematical specialties. Then you can move on to more specialized literature.
It is worth saying that, in addition to differential, there are also integral equations, so you will always have something to strive for and what to study.
Conclusion
We hope that after reading this article you have an idea of what differential equations are and how to solve them correctly.
In any case, mathematics will be useful to us in some way in life. She develops logic and attention, without which each person is as if without hands.