Linear and homogeneous first-order differential equations. Sample solutions

I think we should start with the history of such a glorious mathematical tool as differential equations. Like all differential and integral calculi, these equations were invented by Newton at the end of the 17th century. He considered this very discovery to be so important that he even encrypted the message, which today can be translated like this: "All laws of nature are described by differential equations." It may seem like an exaggeration, but it's true. 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 is now being studied at senior university courses.

A new milestone in the study of differential equations began with Henri Poincare. 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 people are afraid of one phrase "differential equation". However, in this article, we will detail the whole essence of this very useful mathematical apparatus, which in fact is not as complex as it seems from the title. In order to start telling about differential equations of the first order, you should first get acquainted with the basic concepts that are inherently associated with this definition. And we will start with the differential.

Differential

Many people know this concept from school. However, we will dwell on it in more detail. Imagine a function graph. We can increase it to such an extent that any of its segments will take 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) is an infinitesimal. It is called the differential and is denoted by the signs dy (the differential of y) and dx (the differential of x). It is very important to understand that the differential is not a finite quantity, and this is its meaning and basic function.

And now we need to consider the following element, which we need in explaining the concept of a differential equation. This is a derivative.

Derivative

We all probably heard in school and this concept. It is said that the derivative is the rate of growth or decrease of the function. However, much of this definition becomes incomprehensible. Let's try to explain the derivative through the differentials. Let's return to an infinitesimal piece of function with two points, which are at the minimum distance from each other. But even for this distance the function has time to change to some extent. And to describe this change and come up with a derivative that can otherwise be written as the ratio of the differentials: f (x) '= df / dx.

Now we need to consider the basic properties of the derivative. There are only three of them:

1. 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 '.
2. The second property is related to multiplication. The product derivative is the sum of the products of one function on the derivative of the other: (a * b) '= a' * b + a * b '.
3. The derivative of the difference can be written in the form of the following equation: (a / b) '= (a' * ba * b ') / b ² .

All these properties are useful for finding solutions of first-order differential equations.

There are also partial 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 as a constant and simply differentiate.

Integral

Another important concept is the integral. In fact, this is the direct opposite of the derivative. The integrals are of several types, but for solving the simplest differential equations we need the most trivial indefinite integrals.

So, what is an integral? Suppose we have a certain dependence of f on x. We take from it the integral and obtain 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 it is very often necessary to take them to find a solution.

Equations are different depending on their nature. In the next section, we will consider the types of first-order differential equations, and then learn how to solve them.

Classes of differential equations

The "diffusers" are divided according to the order of the derivatives participating in them. Thus there is a first, second, third or more order. They can also be divided into several classes: ordinary and in partial derivatives.

In this paper we consider first-order ordinary differential equations. Examples and methods for solving them will also be discussed in the following sections. We will consider only ODE, because these are the most common types of equations. Ordinary are divided into subspecies: with separating 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 obtain a system of first-order differential equations. We will also consider such systems and learn how to solve them.

Why do we consider only the first order? Because you need to start with a simple one, and it's simply impossible to describe everything related to differential equations in one article.

Equations with separable variables

These are, perhaps, the simplest first-order differential equations. These include examples that can be written as: y '= f (x) * f (y). To solve this equation we need the formula for representing the derivative as the ratio of the differentials: y '= dy / dx. With the help of it we get the following equation: dy / dx = f (x) * f (y). Now we can turn to the method of solving standard examples: we divide the variables by parts, that is, we transfer everything from the variable y to the part where dy is located, and we also do this with the variable x. We obtain an equation of the form dy / f (y) = f (x) dx, which is solved by taking integrals from both sides. Do not forget about the constant, which must be set after taking the integral.

The solution of any "diffuser" 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's analyze on a concrete example the whole course of the solution:

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 "igruk" 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 given, for example, y (n / 2) = e. Then we just substitute the value of these variables in the solution and find the value of the constant. In our example, it is 1.

Homogeneous first-order differential equations

Now go to the more complex part. Homogeneous first-order differential equations can be written in the general form as follows: y '= z (x, y). It should be noted that the right function of two variables is homogeneous, and it can not be divided into two dependencies: z from x and z from y. To check whether the equation is homogeneous or not, it is quite simple: we make the substitution x = k * x and y = k * y. Now we cut all k. If all these letters are reduced, then the equation is homogeneous and you can safely proceed to its solution. Running ahead, let's say: the principle of solving these examples is also very simple.

We need to make a substitution: y = t (x) * x, where t is a 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 the separating variables t and x. We solve it and get the dependence t (x). When we got it, then simply substitute y = t (x) * x in our previous substitution. Then we get the dependence of y on x.

To make it more clear, let's take an example: x * y '= yx * e ^{y / x} .

At check with replacement all is reduced. Hence, the equation is really homogeneous. Now we make another substitution, about which we spoke: 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). It only remains for us to replace t by y / x (because if y = t * x, then t = y / x), and we get the answer: e ^{-y / x} = ln (x * C).

Linear first-order differential equations

It's time to consider another broad topic. We will analyze first-order inhomogeneous differential equations. How do they differ from the previous two? Let's figure it out. Linear differential equations of the first order can be written in general form by the following equation: y '+ g (x) * y = z (x). It is worthwhile to clarify that z (x) and g (x) can be constant quantities.

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 variation of arbitrary constants.

In order to solve the equation in this way, it is first necessary to equate the right-hand side to zero and solve the resulting equation, which after the transfer of the parts takes the form:

Y '= y * x;

Dy / dx = y * x;

Dy / y = xdx;

Ln | y | = x 2/2 + C;

Y = e ^{x2 / 2} * at ^C = C ₁ * e ^{x2 / 2} .

Now we need to replace the constant C ₁ by 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 .

It can be seen that on the left side two terms cancel. If in some example this did not happen, then you did something wrong. Let's continue:

V '* e ^{x2 / 2} = x2.

Now we solve the usual equation, in which we need to separate the variables:

Dv / dx = x ² / e ^{x2 / 2} ;

Dv = x2 * e ^{- x2 / 2} dx.

In order to extract the integral, we have to apply the integration by parts. However, this is not the topic of our article. If you are interested, you can learn how to do it yourself. It is not difficult, and with sufficient skill and attention 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 - it's up to you.

So, when solving the equation by this method, we need to make a substitution: y = k * n. Here k and n are certain functions depending on x. 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 equate to zero what is in brackets. Now, if we combine the two resulting equations, we obtain a system of first-order differential equations, which must be solved:

N '+ x * n = 0;

K '* n = x ² .

The first equation is solved as an ordinary equation. To do this, you need to separate the variables:

Dn / dx = x * v;

Dn / n = xdx.

We take the integral and obtain: ln (n) = x 2/2. Then, if we express n:

N = e ^{x2 / 2} .

Now we substitute the resulting equality into the second equation of the system:

K '* e ^{x2 / 2} = x2.

And transforming, we get the same equality as in the first method:

Dk = x ² / e ^{x2 / 2} .

We also will not disassemble 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 subject, it starts to get better and better.

Where are the differential equations used?

Very active differential equations are used in physics, since almost all the basic laws are written in differential form, and those formulas that we see are the solution of 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 predator-prey. They can also be used to create models of reproduction, say, a colony of microorganisms.

How will differential equations help in life?

The answer to this question is simple: no way. If you are not a scientist or an engineer, they are unlikely to be useful to you. However, for a general development, it is not unreasonable to know what a differential equation is and how it is solved. And then the question of the son or daughter "what is the differential equation?" Do not put you in a cul-de-sac. Well, if you are a scientist or an engineer, you yourself understand the importance of this topic in any science. But the main thing is that now the question "how to solve the differential equation of the first order?" You can always give an answer. Agree, it's always pleasant when you understand what people are even afraid to understand.

The main problems in the study

The main problem in understanding this topic is the poor skill of integrating and differentiating functions. If you do not take derivatives and integrals badly, then probably, it is worthwhile to learn, to master different methods of integration and differentiation, and only then to start studying 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 people do not immediately realize that solving first-order differential equations is often a function or a non-integral integral, and this delusion gives them a lot of trouble.

What else can you study for a better understanding?

It is best to start a further immersion in the world of differential calculus from specialized textbooks, for example, in mathematical analysis for students of non-mathematical specialties. Then you can go to more specialized literature.

It is worth mentioning that, in addition to differential equations, there are also integral equations, so that 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 in any way useful to us in life. It develops the logic and attention, without which every person is like without hands.