Ordinary differential equation is an equation that relates an independent variable, an unknown function of this variable and its derivatives (or differentials) of various orders.
The order of the differential equation is called the order of the highest derivative contained in it.
In addition to ordinary ones, partial differential equations are also studied. These are equations relating independent variables, an unknown function of these variables and its partial derivatives with respect to the same variables. But we will only consider ordinary differential equations and therefore, for the sake of brevity, we will omit the word “ordinary”.
Examples of differential equations:
(1) 
;
(3) 
;
(4) 
;
Equation (1) is fourth order, equation (2) is third order, equations (3) and (4) are second order, equation (5) is first order.
Differential equation n th order does not necessarily have to contain an explicit function, all its derivatives from the first to n-th order and independent variable. It may not contain explicit derivatives of certain orders, a function, or an independent variable.
For example, in equation (1) there are clearly no third- and second-order derivatives, as well as a function; in equation (2) - the second-order derivative and the function; in equation (4) - the independent variable; in equation (5) - functions. Only equation (3) contains explicitly all the derivatives, the function and the independent variable.
Solving a differential equation every function is called y = f(x), when substituted into the equation it turns into an identity.
The process of finding a solution to a differential equation is called its integration.
Example 1. Find the solution to the differential equation.
Solution. Let's write this equation in the form . The solution is to find the function from its derivative. The original function, as is known from integral calculus, is an antiderivative for, i.e.
That's what it is solution to this differential equation . Changing in it C, we will obtain different solutions. We found out that there is an infinite number of solutions to a first order differential equation.
General solution of the differential equation n th order is its solution, expressed explicitly with respect to the unknown function and containing n independent arbitrary constants, i.e.
The solution to the differential equation in Example 1 is general.
Partial solution of the differential equation a solution in which arbitrary constants are given specific numerical values is called.
Example 2. Find the general solution of the differential equation and a particular solution for 
.
Solution. Let's integrate both sides of the equation a number of times equal to the order of the differential equation.
,
.
As a result, we received a general solution -
of a given third order differential equation.
Now let's find a particular solution under the specified conditions. To do this, substitute their values instead of arbitrary coefficients and get
.
If, in addition to the differential equation, the initial condition is given in the form , then such a problem is called Cauchy problem . Substitute the values and into the general solution of the equation and find the value of an arbitrary constant C, and then a particular solution of the equation for the found value C. This is the solution to the Cauchy problem.
Example 3. Solve the Cauchy problem for the differential equation from Example 1 subject to .
Solution. Let us substitute the values from the initial condition into the general solution y = 3, x= 1. We get
We write down the solution to the Cauchy problem for this first-order differential equation:
Solving differential equations, even the simplest ones, requires good integration and derivative skills, including complex functions. This can be seen in the following example.
Example 4. Find the general solution to the differential equation.
Solution. The equation is written in such a form that you can immediately integrate both sides.
.
We apply the method of integration by change of variable (substitution). Let it be then.
Required to take dx and now - attention - we do this according to the rules of differentiation of a complex function, since x and there is a complex function (“apple” is the extraction of a square root or, which is the same thing, raising to the power “one-half”, and “minced meat” is the very expression under the root):
We find the integral:
Returning to the variable x, we get:
.
This is the general solution to this first degree differential equation.
Not only skills from previous sections of higher mathematics will be required in solving differential equations, but also skills from elementary, that is, school mathematics. As already mentioned, in a differential equation of any order there may not be an independent variable, that is, a variable x. Knowledge about proportions from school that has not been forgotten (however, depending on who) from school will help solve this problem. This is the next example.
This article is a starting point in studying the theory of differential equations. Here are the basic definitions and concepts that will constantly appear in the text. For better assimilation and understanding, the definitions are provided with examples.
Differential equation (DE) is an equation that includes an unknown function under the derivative or differential sign.
If the unknown function is a function of one variable, then the differential equation is called ordinary(abbreviated ODE - ordinary differential equation). If the unknown function is a function of many variables, then the differential equation is called partial differential equation.
The maximum order of the derivative of an unknown function entering a differential equation is called order of the differential equation.
Here are examples of ODEs of the first, second and fifth orders, respectively
As examples of second order partial differential equations, we give 
Further we will consider only ordinary differential equations of the nth order of the form 
or 
, where Ф(x, y) = 0 is an unknown function specified implicitly (when possible, we will write it in explicit representation y = f(x) ).
The process of finding solutions to a differential equation is called by integrating the differential equation.
Solving a differential equation is an implicitly specified function Ф(x, y) = 0 (in some cases, the function y can be expressed explicitly through the argument x), which turns the differential equation into an identity.
NOTE.
The solution to a differential equation is always sought on a predetermined interval X.
Why are we talking about this separately? Yes, because in many problems the interval X is not mentioned. That is, usually the condition of the problems is formulated as follows: “find a solution to the ordinary differential equation " In this case, it is implied that the solution should be sought for all x for which both the desired function y and the original equation make sense.
The solution to a differential equation is often called integral of the differential equation.
Functions or can be called the solution of a differential equation.
One of the solutions to the differential equation is the function. Indeed, substituting this function into the original equation, we obtain the identity 
. It is easy to see that another solution to this ODE is, for example, . Thus, differential equations can have many solutions.
General solution of a differential equation is a set of solutions containing all, without exception, solutions to this differential equation.
The general solution of a differential equation is also called general integral of the differential equation.
Let's go back to the example. The general solution of the differential equation has the form or , where C is an arbitrary constant. Above we indicated two solutions to this ODE, which are obtained from the general integral of the differential equation by substituting C = 0 and C = 1, respectively.
If the solution of a differential equation satisfies the initially specified additional conditions, then it is called partial solution of the differential equation.
A partial solution of the differential equation satisfying the condition y(1)=1 is . Really, 
And 
.
The main problems of the theory of differential equations are Cauchy problems, boundary value problems and problems of finding a general solution to a differential equation on any given interval X.
Cauchy problem is the problem of finding a particular solution to a differential equation that satisfies the given initial conditions, where are numbers.
Boundary value problem is the problem of finding a particular solution to a second-order differential equation that satisfies additional conditions at the boundary points x 0 and x 1:
f (x 0) = f 0, f (x 1) = f 1, where f 0 and f 1 are given numbers.
The boundary value problem is often called boundary problem.
An ordinary differential equation of nth order is called linear, if it has the form , and the coefficients are continuous functions of the argument x on the integration interval.
Differential equation (DE)
- this is the equation,
where are the independent variables, y is the function and are the partial derivatives.
Ordinary differential equation is a differential equation that has only one independent variable, .
Partial differential equation is a differential equation that has two or more independent variables.
The words “ordinary” and “partial derivatives” may be omitted if it is clear which equation is being considered. In what follows, ordinary differential equations are considered.
Order of differential equation is the order of the highest derivative.
Here is an example of a first order equation:
Here is an example of a fourth order equation:
Sometimes a first order differential equation is written in terms of differentials:
In this case, the variables x and y are equal. That is, the independent variable can be either x or y. In the first case, y is a function of x. In the second case, x is a function of y. If necessary, we can reduce this equation to a form that explicitly includes the derivative y′.
Dividing this equation by dx we get:
.
Since and , it follows that
.
Solving differential equations
Derivatives of elementary functions are expressed through elementary functions. Integrals of elementary functions are often not expressed in terms of elementary functions. With differential equations the situation is even worse. As a result of the solution you can get:
- explicit dependence of a function on a variable;
Solving a differential equation is the function y = u (x), which is defined, n times differentiable, and .
- implicit dependence in the form of an equation of type Φ (x, y) = 0 or systems of equations;
Integral of a differential equation is a solution to a differential equation that has an implicit form.
- dependence expressed through elementary functions and integrals from them;
Solving a differential equation in quadratures - this is finding a solution in the form of a combination of elementary functions and integrals of them.
- the solution may not be expressed through elementary functions.
Since solving differential equations comes down to calculating integrals, the solution includes a set of constants C 1, C 2, C 3, ... C n. The number of constants is equal to the order of the equation. Partial integral of a differential equation is the general integral for given values of the constants C 1, C 2, C 3, ..., C n.
References:
V.V. Stepanov, Course of differential equations, "LKI", 2015.
N.M. Gunther, R.O. Kuzmin, Collection of problems in higher mathematics, “Lan”, 2003.
General integral ordinary differential equation F (x, y, y",..., y (n)) =0 Ratio Φ( x, y, C 1 ,..., C n) =0, containing also essential arbitrary constants C 1 ,..., C n , the consequence of which is this differential equation (see Differential equations). In other words, this equation must be the result of eliminating the constants C 1 (i = 1,..., n) from the equations: Moreover, these constants are essential in the sense that the process of eliminating them from the system (*) cannot lead to a differential equation different from the given one. O. and. is closely related to the general solution (See General solution). If permanent C i, included in the O. and., give certain values, then we obtain a frequent integral. Incomplete elimination of constants C i from system (*) leads to the intermediate integral Fk(x, y, y",..., y (n-k)), C 1,..., C k = 0 (where 1 ≤ k ≤n- 1); in particular, for k = 1 - to the first integral (See First integral). Geometrically O. and. is n-parametric family of integral curves.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what “General integral” is in other dictionaries:
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When solving various problems in physics, chemistry, mathematics and other exact sciences, mathematical models are often used in the form of equations relating one or more independent variables, an unknown function of these variables and derivatives (or differentials) of this function. This kind the equations are called differential.
If there is only one independent variable, then the equation is called ordinary; if there are two or more independent variables, then the equation is called partial differential equation. In order to obtain highly qualified specialists, a course in differential equations is required in all universities where exact disciplines are studied. For some students, theory is difficult, practice is a struggle; for others, both theory and practice are difficult. If you analyze differential equations from a practical perspective, then to calculate them you only need to be good at integrating and taking derivatives. All other transformations come down to several schemes that can be understood and studied. Below we will study the basic definitions and method for solving simple DR.
Theory of differential equations
Definition: Ordinary differential equation is an equation that connects the independent variable x, the function y(x), its derivatives y"(x), y n (x) and has the general form F(x,y(x),y" (x), …, y n (x))=0
Differential equation(DR) is called either an ordinary differential equation or a partial differential equation. Order of differential equation is determined by the order of the highest derivative (n), which is included in this differential equation.
General solution of the differential equation is a function that contains as many constants as the order of the differential equation, and the substitution of which into a given differential equation turns it into an identity, that is, it has the form y=f(x, C 1, C 2, ..., C n).
A general solution that is not resolved with respect to y(x) and has the form F(x,y,C 1 ,C 2 , …, C n)=0 is called general integral of a differential equation.
The solution found from the general one for fixed values of the constants C 1 , C 2 , …, C n is called private solution of a differential equation.
The simultaneous specification of a differential equation and the corresponding number of initial conditions is called Cauchy problem.
F(x,y,C 1 ,C 2 , …, C n)=0
y(x0)=y0;
….
y n (x0)=y n (0)
Ordinary differential equation of the first order called an equation of the form
F(x, y, y")=0. (1)
Integral of the equation(1) is called a relation of the form Ф (x,y)=0 if each continuously differentiated function implicitly specified by it is a solution to equation (1).
An equation that has the form (1) and cannot be reduced to a simple form is called an equation, undecidable with respect to the derivative. If it can be written in the form
y" = f(x,y), then it is called solved equation for the derivative.
Cauchy problem for a first order equation contains only one initial condition and has the form:
F(x,y,y")=0
y(x 0)=y 0 .
Equations of the form
M(x,y)dx+N(x,y)dx=0 (2)
where the variables x i y are "symmetric": we can assume that x is an independent variable and y is a dependent variable, or vice versa, y is an independent variable and x is a dependent variable, called equation in symmetric form.
Geometric meaning of a first order differential equation
y"=f(x,y) (3)
is as follows.
This equation establishes a connection (dependence) between the coordinates of the point (x;y) and the angular coefficient y" of the tangent to the integral curve passing through this point. Thus, the equation y"= f(x,y) is a set directions (directions field) on the Cartesian Oxy plane.
A curve constructed at points at which the direction of the field is the same is called an isocline. Isoclins can be used to approximate the construction of integral curves. The isocline equation can be obtained by putting the derivative equal to the constant y"=C
f(x, y)=C - isocline equation..
Integral line of the equation(3) is called the graph of the solution to this equation.
Ordinary differential equations whose solutions can be specified analytically y=g(x) are called integrable equations.
Equations of the form
M 0 (x)dx+N 0 (y)dy=0 (3)
are called equations with separate interchangeables.
From them we will begin our acquaintance with differential equations. The process of finding solutions to DR is called integration of a differential equation.
Separated Variable Equations
Example 1. Find the solution to the equation y"=x .
Check the solution.
Solution: Write the equation in differentials
dy/dx=x or dy=x*dx.
Let's find the integral of the right and left sides of the equation
int(dy)=int(x*dx);
y=x 2 /2+C.
This is the DR integral.
Let's check its correctness and calculate the derivative of the function
y"=1/2*2x+0=x.
As you can see, we received the original DR, therefore the calculations are correct.
We have just found a solution to a first order differential equation. This is exactly the simplest equation you can imagine.
Example 2. Find the general integral of a differential equation
(x+1)y"=y+3
Solution: Let's write the original equation in differentials
(x+1)dy=(y+3)dx.
The resulting equation is reduced to DR with separated variables
All that's left is to take the integral of both sides 
Using tabular formulas we find
ln|y+3|=ln|x+1|+C.
If we expose both parts, we get
y+3=e ln|x+1|+C or y=e ln|x+1|+C -3.
This notation is correct, but not compact.
In practice, a different technique is used; when calculating the integral, the constant is entered under the logarithm
ln|y+3|=ln|x+1|+ln(C).
According to the properties of the logarithm, this allows you to collapse the last two terms
ln|y+3|=ln(С|x+1|).
Now when exposing solving a differential equation will be compact and easy to read
y=С|x+1|+3
Remember this rule, in practice it is used as a calculation standard.
Example 3. Solve differential equation
y"=-y*sin(x).
Solution: Let's write it down equation in differentials
dy/dx= y*sin(x)
or after rearranging the factors in the form separated equations
dy/ y=-sin(x)dx.
It remains to integrate the equation
int(1/y,y)=-int(sin(x), x);
ln|y|=cos(x)-ln(C).
It is convenient to enter the constant under the logarithm, and even with a negative value, in order to transfer it to the left side to obtain
ln|С*y|=cos(x).
Exposing both sides of the dependence
С*y=exp(cos(x)).
This is it. You can leave it as is, or you can permanently move it to the right side.
The calculations are not complicated; in most cases, integrals can also be found using tabular integration formulas.
Example 4. Solve the Cauchy problem
y"=y+x, y(1)=e 3 -2.
Solution: Here the preliminary transformations will no longer take place. However, the equation is linear and quite simple. In such cases, you need to introduce a new variable
z=y+x.
Remembering that y=y(x) we find the derivative of z.
z"= y"+1,
from where we express the old derivative
y"= z"-1.
Let's substitute all this into the original equation
z"-1=z or z"=z+1.
Let's write it down differential equation through differentials
dz=(z+1)dx.
Separating the variables in the equation
All that remains is to calculate simple integrals that anyone can do 
We expose the dependence to get rid of the logarithm of the function
z+1=e x+C or z=e x+1 -1
Don't forget to return to the completed replacement.
z=x+y= e x+С -1,
write it out from here general solution to differential equation
y= e x+C -x-1.
Finding a solution to the Cauchy problem in DR in this case is not difficult. We write out the Cauchy condition
y(1)=e 3 -2
and substitute into the solution we just found
e 1 + C -1-1 = e 3 -2.
From here we obtain the condition for calculating the constant
1+C=3; C=3-1=2.
Now we can write solution of the Cauchy problem (partial solution of DR)
y= e x+2 -x-1.
If you know how to integrate well, and you are also doing well with derivatives, then the topic of differential equations will not be an obstacle in your education.
In further study, you will need to study several important diagrams so that you can distinguish between equations and know which substitution or technique works in each case.
After this, homogeneous and inhomogeneous DRs, differential equations of the first and higher orders await you. In order not to burden you with theory, in the following lessons we will give only the type of equations and a brief scheme for their calculations. You can read the whole theory from methodological recommendations for studying the course "Differential equations"(2014) authors Bokalo Nikolay Mikhailovich, Domanskaya Elena Viktorovna, Chmyr Oksana Yuryevna. You can use other sources that contain explanations of the theory of differential equations that you understand. Ready-made examples for differential. equations taken from the program for mathematicians of LNU. I. Frank.
We know how to solve differential equations and will try to instill this knowledge in you in an easy way.
