Numerical solution of 2nd order differential equations. Numerical solution of ordinary differential equations

Laboratory work 1

Numerical solution methods

ordinary differential equations(4 hours)

When solving many physical and geometric problems one has to look for an unknown function based on a given relationship between the unknown function, its derivatives and independent variables. This ratio is called differential equation , and finding a function that satisfies the differential equation is called solving a differential equation.

Ordinary differential equation called equality

, (1)

in which

is an independent variable that changes in a certain segment, and - unknown function y ( x ) and her first n derivatives. called order of the equation .

The task is to find a function y that satisfies equality (1). Moreover, without stipulating this separately, we will assume that the desired solution has one or another degree of smoothness necessary for the construction and “legal” application of one or another method.

There are two types of ordinary differential equations

Equations without initial conditions

Equations with initial conditions.

Equations without initial conditions are equations of the form (1).

Equation with initial conditions is an equation of the form (1), in which it is required to find such a function

, which for some satisfies the following conditions: ,

those. at the point

the function and its first derivatives take on predetermined values.

Cauchy problems

When studying methods for solving differential equations using approximate methods main task counts Cauchy problem.

Let's consider the most popular method for solving the Cauchy problem - the Runge-Kutta method. This method allows you to construct formulas for calculating an approximate solution of almost any order of accuracy.

Let us derive the formulas of the Runge-Kutta method of second order accuracy. To do this, we represent the solution as a piece of a Taylor series, discarding terms with an order higher than the second. Then the approximate value of the desired function at the point x 1 can be written as:

(2)

Second derivative y "( x 0 ) can be expressed through the derivative of the function f ( x , y ) , however, in the Runge-Kutta method, instead of the derivative, the difference is used

selecting parameter values ​​accordingly

Then (2) can be rewritten as:

y 1 = y 0 + h [ β f ( x 0 , y 0 ) + α f ( x 0 + γh , y 0 + δh )], (3)

Where α , β , γ And δ – some parameters.

Considering the right-hand side of (3) as a function of the argument h , let's break it down into degrees h :

y 1 = y 0 +( α + β ) h f ( x 0 , y 0 ) + αh 2 [ γ f x ( x 0 , y 0 ) + δ f y ( x 0 , y 0 )],

and select the parameters α , β , γ And δ so that this expansion is close to (2). It follows that

α + β =1, αγ =0,5, α δ =0,5 f ( x 0 , y 0 ).

Using these equations we express β , γ And δ via parameters α , we get

y 1 = y 0 + h [(1 - α ) f ( x 0 , y 0 ) + α f ( x 0 +, y 0 + f ( x 0 , y 0 )], (4)

0 < α ≤ 1.

Now, if instead of ( x 0 , y 0 ) in (4) substitute ( x 1 , y 1 ), we get a formula for calculating y 2 – approximate value of the desired function at the point x 2 .

In the general case, the Runge-Kutta method is applied to an arbitrary partition of the segment [ x 0 , X ] on n parts, i.e. with variable pitch

x 0 , x 1 , …, x n ; h i = x i+1 – x i , x n = X. (5)

Options α are chosen equal to 1 or 0.5. Let us finally write down the calculation formulas of the second order Runge-Kutta method with variable steps for α =1:

y i+1 =y i +h i f(x i + , y i + f(x i , y i)), (6.1)

i = 0, 1,…, n -1.

And α =0,5:

y i+1 =y i + , (6.2)

i = 0, 1,…, n -1.

The most used formulas of the Runge-Kutta method are formulas of the fourth order of accuracy:

y i+1 =y i + (k 1 + 2k 2 + 2k 3 + k 4),

k 1 =f(x i , y i), k 2 = f(x i + , y i + k 1), (7)

k 3 = f(x i + , y i + k 2), k 4 = f(x i +h, y i +hk 3).

For the Runge-Kutta method, Runge's rule is applicable to estimate the error. Let y ( x ; h ) – approximate value of the solution at the point x , obtained by formulas (6.1), (6.2) or (7) with step h , A p – the order of accuracy of the corresponding formula. Then the error R ( h ) values y ( x ; h ) can be estimated using an approximate value y ( x ; 2 h ) solutions at a point x , obtained in increments 2 h :

(8)

Where p =2 for formulas (6.1) and (6.2) and p =4 for (7).

We consider only the solution to the Cauchy problem. A system of differential equations or one equation must be converted to the form

Where ,
–n-dimensional vectors; y– unknown vector function; x– independent argument,
. In particular, if n= 1, then the system turns into one differential equation. The initial conditions are set as follows:
, Where
.

If
in the vicinity of a point
is continuous and has continuous partial derivatives with respect to y, then the existence and uniqueness theorem guarantees that there is only one continuous vector function
, defined in some neighborhood of a point , satisfying equation (7) and the condition
.

Let us pay attention to the fact that the neighborhood of the point , where the solution is determined, can be very small. When approaching the boundary of this neighborhood, the solution can go to infinity, oscillate, with an infinitely increasing frequency, in general, behave so badly that it cannot be continued beyond the boundary of the neighborhood. Accordingly, such a solution cannot be tracked by numerical methods on a larger segment, if one is specified in the problem statement.

Solving the Cauchy problem on [ a; b] is a function. In numerical methods, the function is replaced by a table (Table 1).

Table 1

Here
,
. The distance between adjacent table nodes is usually taken to be constant:
,
.

There are tables with variable steps. The table step is determined by the requirements of the engineering problem and not connected with the accuracy of finding a solution.

If y is a vector, then the table of solution values ​​will take the form of a table. 2.

Table 2

In the MATHCAD system, a matrix is ​​used instead of a table, and it is transposed with respect to the specified table.

Solve the Cauchy problem with accuracy ε means to get the values ​​in the specified table (numbers or vectors),
, such that
, Where
- exact solution. It is possible that the solution to the segment specified in the problem does not continue. Then you need to answer that the problem cannot be solved on the entire segment, and you need to get a solution on the segment where it exists, making this segment as large as possible.

It should be remembered that the exact solution
we don’t know (otherwise why use the numerical method?). Grade
must be justified on some other basis. As a rule, it is not possible to obtain a 100% guarantee that the assessment is being carried out. Therefore, algorithms are used to estimate the value
, which prove effective in most engineering problems.

The general principle for solving the Cauchy problem is as follows. Line segment [ a; b] is divided into a number of segments by integration nodes. Number of nodes k does not have to match the number of nodes m final table of decision values ​​(Tables 1, 2). Usually, k > m. For simplicity, we will assume that the distance between nodes is constant,
;h called the integration step. Then, according to certain algorithms, knowing the values at i < s, calculate the value . The smaller the step h, the lower the value will differ from the value of the exact solution
. Step h in this division is already determined not by the requirements of the engineering problem, but by the required accuracy of solving the Cauchy problem. In addition, it must be selected so that at one step the table. 1, 2 fit an integer number of steps h. In this case the values y, obtained as a result of calculations with steps h at points
, are used accordingly in the table. 1 or 2.

The simplest algorithm for solving the Cauchy problem for equation (7) is the Euler method. The calculation formula is:

(8)

Let's see how the accuracy of the solution found is assessed. Let's pretend that
is the exact solution of the Cauchy problem, and also that
, although this is almost always not the case. Then where is the constant C depends on function
in the vicinity of a point
. Thus, at one step of integration (finding a solution) we get an error of the order . Because steps have to be taken
, then it is natural to expect that the total error at the last point
everything will be fine
, i.e. order h. Therefore, Euler's method is called the first order method, i.e. the error has the order of the first power of the step h. In fact, at one step of integration the following estimate can be justified. Let
– exact solution of the Cauchy problem with the initial condition
. It's clear that
does not coincide with the required exact solution
the original Cauchy problem of equation (7). However, at small h and "good" function
these two exact solutions will differ little. The Taylor remainder formula ensures that
, this gives the integration step error. The final error consists not only of errors at each integration step, but also of deviations of the desired exact solution
from exact solutions
,
, and these deviations can become very large. However, the final estimate of the error in the Euler method for a “good” function
still looks like
,
.

When applying Euler's method, the calculation proceeds as follows. According to specified accuracy ε determine the approximate step
. Determining the number of steps
and again approximately select the step
. Then again we adjust it downward so that at each step the table. 1 or 2 fit an integer number of integration steps. We get a step h. According to formula (8), knowing And , we find. By found value And
we find so on.

The resulting result may not, and generally will not, have the desired accuracy. Therefore, we reduce the step by half and again apply the Euler method. We compare the results of the first application of the method and the second in identical points . If all discrepancies are less than the specified accuracy, then the last calculation result can be considered the answer to the problem. If not, then we reduce the step by half again and apply Euler’s method again. Now we compare the results of the last and penultimate application of the method, etc.

Euler's method is used relatively rarely due to the fact that to achieve a given accuracy ε a large number of steps are required, in the order of
. However, if
has discontinuities or discontinuous derivatives, then higher order methods will produce the same error as Euler's method. That is, the same amount of calculations will be required as in the Euler method.

Of the higher order methods, the fourth order Runge–Kutta method is most often used. In it, calculations are carried out according to the formulas

This method, in the presence of continuous fourth derivatives of the function
gives an error on one step of the order , i.e. in the notation introduced above,
. In general, on the integration interval, provided that the exact solution is determined on this interval, the integration error will be of the order of .

The selection of the integration step occurs in the same way as described in Euler’s method, except that the initial approximate value of the step is selected from the relation
, i.e.
.

Most programs used to solve differential equations use automatic step selection. The gist of it is this. Let the value already be calculated . The value is calculated
in increments h, chosen during calculation . Then two integration steps are performed with step , i.e. extra node is added
in the middle between the nodes And
. Two values ​​are calculated
And
in nodes
And
. The value is calculated
, Where p– method order. If δ is less than the accuracy specified by the user, then it is assumed
. If not, then choose a new step h equal and repeat the accuracy check. If during the first check δ is much less than the specified accuracy, then an attempt is made to increase the step. For this purpose it is calculated
at the node
in increments h from node
and is calculated
in steps of 2 h from node . The value is calculated
. If is less than the specified accuracy, then step 2 h considered acceptable. In this case, a new step is assigned
,
,
. If more accuracy, then the step is left the same.

It should be taken into account that programs with automatic selection of the integration step achieve the specified accuracy only when performing one step. This occurs due to the accuracy of the approximation of the solution passing through the point
, i.e. approximation of the solution
. Such programs do not take into account how much the solution
differs from the desired solution
. Therefore, there is no guarantee that the specified accuracy will be achieved throughout the entire integration interval.

The described Euler and Runge–Kutta methods belong to the group of one-step methods. This means that to calculate
at the point
it is enough to know the meaning at the node . It is natural to expect that if more information about the decision is used, several previous values ​​of it will be taken into account
,
etc., then the new value
it will be possible to find more accurately. This strategy is used in multi-step methods. To describe them, we introduce the notation
.

Representatives of multi-step methods are the Adams–Bashforth methods:

Method k-th order gives a local order error
or global – order .

These methods belong to the group of extrapolation methods, i.e. the new meaning is clearly expressed through the previous ones. Another type is interpolation methods. In them, at each step, you have to solve a nonlinear equation for a new value . Let's take the Adams–Moulton methods as an example:

To use these methods, you need to know several values ​​at the beginning of the count
(their number depends on the order of the method). These values ​​must be obtained by other methods, for example the Runge–Kutta method with a small step (to increase accuracy). Interpolation methods in many cases turn out to be more stable and allow larger steps to be taken than extrapolation methods.

In order not to solve a nonlinear equation at each step in interpolation methods, Adams predictor-correction methods are used. The bottom line is that the extrapolation method is first applied at the step and the resulting value
is substituted into the right side of the interpolation method. For example, in the second order method

To solve differential equations, it is necessary to know the value of the dependent variable and its derivatives for certain values ​​of the independent variable. If additional conditions are specified for one value of the unknown, i.e. independent variable., then such a problem is called the Cauchy problem. If the initial conditions are specified for two or more values ​​of the independent variable, then the problem is called a boundary value problem. When solving differential equations of various types, the function whose values ​​need to be determined is calculated in the form of a table.

Classification of numerical methods for solving differentials. Lv. Types.

Cauchy problem – one-step: Euler methods, Runge-Kutta methods; – multi-step: Main method, Adams method. Boundary problem – a method of reducing a boundary problem to the Cauchy problem; – finite difference method.

When solving the Cauchy problem, dif. must be specified. ur. order n or system of dif. ur. first order of n equations and n additional conditions for its solution. Additional conditions must be specified for the same value of the independent variable. When solving a boundary problem, equations must be specified. nth order or a system of n equations and n additional conditions for two or more values ​​of the independent variable. When solving the Cauchy problem, the required function is determined discretely in the form of a table with a certain specified step . When determining each successive value, you can use information about one previous point. In this case, the methods are called one-step, or you can use information about several previous points - multi-step methods.

Ordinary differential equations. Cauchy problem. One-step methods. Euler's method.

Given: g(x,y)y+h(x,y)=0, y=-h(x,y)/g(x,y)= f(x,y), x 0 , y( x 0)=y 0 . It is known: f(x,y), x 0 , y 0 . Determine the discrete solution: x i , y i , i=0,1,…,n. Euler's method is based on the expansion of a function into a Taylor series in the vicinity of the point x 0 . The neighborhood is described by step h. y(x 0 +h)y(x 0)+hy(x 0)+…+ (1). Euler's method takes into account only two terms of the Taylor series. Let us introduce some notation. Euler's formula will take the form: y i+1 =y i +y i, y i =hy(x i)=hf(x i,y i), y i+1 =y i +hf(x i,y i) (2), i= 0,1,2…, x i+1 =x i +h

Formula (2) is the formula of the simple Euler method.

Geometric interpretation of Euler's formula

For getting numerical solution the tangent line passing through the equation is used. tangent: y=y(x 0)+y(x 0)(x-x 0), x=x 1,

y 1 =y(x 0)+f(x 0 ,y 0)  (x-x 0), because

x-x 0 =h, then y 1 =y 0 +hf(x 0 ,y 0), f(x 0 ,y 0)=tg £.

Modified Euler method

Given: y=f(x,y), y(x 0)=y 0 . It is known: f(x,y), x 0 , y 0 . Determine: the dependence of y on x in the form of a tabular discrete function: x i, y i, i=0.1,…,n.

Geometric interpretation

1) calculate the tangent of the angle of inclination at the starting point

tg £=y(x n ,y n)=f(x n ,y n)

2) Calculate the value  y n+1 on

end of step according to Euler's formula

 y n+1 =y n +f(x n ,y n) 3) Calculate the tangent of the angle of inclination

tangent at n+1 point: tg £=y(x n+1 ,  y n+1)=f(x n+1 ,  y n+1) 4) Calculate the arithmetic mean of the angles

tilt: tg £=½. 5) Using the tangent of the slope angle, we recalculate the value of the function at n+1 point: y n+1 =y n +htg £= y n +½h=y n +½h – formula of the modified Euler method. It can be shown that the resulting f-la corresponds to the expansion of the f-i in a Taylor series, including terms (up to h 2). The modified Eilnra method, unlike the simple one, is a method of second order accuracy, because the error is proportional to h 2.

Numerical solution of differential equations

Many problems in science and technology come down to solving ordinary differential equations (ODEs). ODEs are those equations that contain one or more derivatives of the desired function. In general, the ODE can be written as follows:

Where x is an independent variable, is the i-th derivative of the desired function. n is the order of the equation. The general solution of an nth order ODE contains n arbitrary constants, i.e. the general solution has the form .

To select a single solution, it is necessary to set n additional conditions. Depending on the method of specifying additional conditions, there are two different types of problems: the Cauchy problem and the boundary value problem. If additional conditions are specified at one point, then such a problem is called the Cauchy problem. Additional conditions in the Cauchy problem are called initial conditions. If additional conditions are specified at more than one point, i.e. for different values ​​of the independent variable, then such a problem is called a boundary value problem. The additional conditions themselves are called boundary or boundary conditions.

It is clear that when n=1 we can only talk about the Cauchy problem.

Examples of setting up the Cauchy problem:

Examples of boundary value problems:

It is possible to solve such problems analytically only for some special types of equations.

Numerical methods for solving the Cauchy problem for first-order ODEs

Formulation of the problem. Find a solution to the first order ODE

On the segment provided

When finding an approximate solution, we will assume that the calculations are carried out with a calculated step, the calculation nodes are the interval points [ x 0 , x n ].

The goal is to build a table

x i				x n
y i				y n

those. Approximate values ​​of y are sought at grid nodes.

Integrating the equation on the interval, we obtain

A completely natural (but not the only) way to obtain a numerical solution is to replace the integral in it with some quadrature formula of numerical integration. If we use the simplest formula for left rectangles of the first order

,

then we get explicit Euler formula:

Payment procedure:

Knowing, we find, then etc.

Geometric interpretation of Euler's method:

Taking advantage of what is at the point x 0 the solution is known y(x 0)= y 0 and the value of its derivative, we can write the equation of the tangent to the graph of the desired function at the point:. With a small enough step h the ordinate of this tangent, obtained by substituting into the right side of the value, should differ little from the ordinate y(x 1) solutions y(x) Cauchy problems. Therefore, the point of intersection of the tangent with the line x = x 1 can be approximately taken as the new starting point. Through this point we again draw a straight line, which approximately reflects the behavior of the tangent to at the point. Substituting here (i.e. the intersection with the line x = x 2), we obtain an approximate value y(x) at point x 2: etc. As a result for i-th point we obtain Euler’s formula.

The explicit Euler method has first order accuracy or approximation.

If you use the right rectangle formula: , then we come to the method

This method is called implicit Euler method, since calculating an unknown value from a known value requires solving an equation that is generally nonlinear.

The implicit Euler method has first order accuracy or approximation.

In this method, the calculation consists of two stages:

This scheme is also called the predictor-corrector method (predictive-correcting). In the first stage, the approximate value is predicted with low accuracy (h), and in the second stage this prediction is corrected so that the resulting value has second order accuracy.

Runge–Kutta methods: the idea of ​​constructing explicit Runge–Kutta methods p-th order is to obtain approximations to the values y(x i+1) according to a formula of the form

…………………………………………….

Here a n ,b nj , p n, – some fixed numbers (parameters).

When constructing the Runge–Kutta methods, the parameters of the function ( a n ,b nj , p n) are selected in such a way as to obtain the desired order of approximation.

Runge–Kutta scheme of fourth order of accuracy:

Example. Solve the Cauchy problem:

Consider three methods: explicit Euler method, modified Euler method, Runge–Kutta method.

Exact solution:

Calculation formulas using the explicit Euler method for this example:

Calculation formulas of the modified Euler method:

Calculation formulas for the Runge–Kutta method:

y1 – Euler’s method, y2 – modified Euler’s method, y3 – Runge Kutta’s method.

It can be seen that the most accurate is the Runge–Kutta method.

Numerical methods for solving systems of first-order ODEs

The methods considered can also be used to solve systems of first-order differential equations.

Let us show this for the case of a system of two first-order equations:

Explicit Euler method:

Modified Euler method:

Runge–Kutta scheme of fourth order of accuracy:

Cauchy problems for higher order equations are also reduced to solving systems of ODE equations. For example, consider Cauchy problem for a second order equation

Let's introduce a second unknown function. Then the Cauchy problem is replaced by the following:

Those. in terms of the previous problem: .

Example. Find a solution to the Cauchy problem:

On the segment.

Exact solution:

Really:

Let's solve the problem using the explicit Euler method, modified by the Euler and Runge-Kutta method with step h=0.2.

Let's introduce the function.

Then we obtain the following Cauchy problem for a system of two first-order ODEs:

Explicit Euler method:

Modified Euler method:

Runge–Kutta method:

Euler circuit:

Modified Euler method:

Runge - Kutta scheme:

Max(y-y theory)=4*10 -5

Finite difference method for solving boundary value problems for ODE

Formulation of the problem: find a solution to a linear differential equation

satisfying the boundary conditions:. (2)

Theorem. Let . Then there is a unique solution to the problem.

This problem reduces, for example, to the problem of determining the deflections of a beam that is hinged at its ends.

Main stages of the finite difference method:

1) the area of ​​continuous change of argument () is replaced by a discrete set of points called nodes: .

2) The desired function of the continuous argument x is approximately replaced by the function of the discrete argument on a given grid, i.e. . The function is called a grid function.

3) The original differential equation is replaced by a difference equation with respect to the grid function. This replacement is called difference approximation.

Thus, solving a differential equation comes down to finding the values ​​of the grid function at grid nodes, which are found from solving algebraic equations.

Approximation of derivatives.

To approximate (replace) the first derivative, you can use the formulas:

- right difference derivative,

- left difference derivative,

Central difference derivative.

that is, there are many possible ways to approximate the derivative.

All these definitions follow from the concept of derivative as a limit: .

Based on the difference approximation of the first derivative, we can construct a difference approximation of the second derivative:

Similarly, we can obtain approximations of higher order derivatives.

Definition. The approximation error of the nth derivative is the difference: .

To determine the order of approximation, Taylor series expansion is used.

Let us consider the right-hand difference approximation of the first derivative:

Those. the right difference derivative has first by h order of approximation.

The same is true for the left difference derivative.

The central difference derivative has second order approximation.

The approximation of the second derivative according to formula (3) also has a second order of approximation.

In order to approximate a differential equation, it is necessary to replace all its derivatives with their approximations. Let's consider problem (1), (2) and replace the derivatives in (1):

As a result we get:

(4)

The order of approximation of the original problem is 2, because the second and first derivatives are replaced with order 2, and the rest - exactly.

So, instead of differential equations (1), (2), we obtain the system linear equations for determination at grid nodes.

The diagram can be represented as:

i.e., we got a system of linear equations with a matrix:

This matrix is ​​tridiagonal, i.e. all elements that are not located on the main diagonal and the two diagonals adjacent to it are equal to zero.

By solving the resulting system of equations, we obtain a solution to the original problem.

Differential equations are equations in which an unknown function appears under the derivative sign. The main task of the theory of differential equations is the study of functions that are solutions to such equations.

Differential equations can be divided into ordinary differential equations, in which the unknown functions are functions of one variable, and partial differential equations, in which the unknown functions are functions of two and more variables.

The theory of partial differential equations is more complex and is covered in more complete or specialized mathematics courses.

Let's start studying differential equations with the simplest equation - a first-order equation.

Equation of the form

F(x,y,y") = 0,(1)

where x is an independent variable; y - the required function; y" - its derivative, is called a first-order differential equation.

If equation (1) can be resolved with respect to y", then it takes the form

and is called a first-order equation resolved with respect to the derivative.

In some cases, it is convenient to write equation (2) in the form f (x, y) dx - dy = 0, which is a special case of the more general equation

P(x,y)dx+Q(x,y)dy=O,(3)

where P(x,y) and Q(x,y) are known functions. The equation in symmetric form (3) is convenient because the variables x and y in it are equal, that is, each of them can be considered as a function of the other.

Let us give two basic definitions of the general and particular solutions of the equation.

A general solution to equation (2) in a certain region G of the Oxy plane is a function y = μ(x,C), depending on x and an arbitrary constant C, if it is a solution to equation (2) for any value of the constant C, and if for any initial conditions y x=x0 =y 0 such that (x 0 ;y 0)=G, there is a unique value of the constant C = C 0 such that the function y=q(x,C 0) satisfies the given initial conditions y=q(x 0 ,C).

A particular solution to equation (2) in the domain G is the function y=ts(x,C 0), which is obtained from the general solution y=ts(x,C) at a certain value of the constant C=C 0.

Geometrically, the general solution y = μ (x, C) is a family of integral curves on the Oxy plane, depending on one arbitrary constant C, and the particular solution y = μ (x, C 0) is one integral curve of this family passing through a given point (x 0; y 0).

Approximate solution of first order differential equations by Euler's method. The essence of this method is that the desired integral curve, which is a graph of a particular solution, is approximately replaced by a broken line. Let the differential equation be given

and initial conditions y |x=x0 =y 0 .

Let us find an approximate solution to the equation on the interval [x 0 ,b] that satisfies the given initial conditions.

Let's divide the segment [x 0 ,b] with points x 0<х 1 ,<х 2 <...<х n =b на n равных частей. Пусть х 1 --х 0 =х 2 -- x 1 = ... =x n -- x n-1 = ?x. Обозначим через y i приближенные значения искомого решения в точках х i (i=1, 2, ..., n). Проведем через точки разбиения х i - прямые, параллельные оси Оу, и последовательно проделаем следующие однотипные операции.

Let's substitute the values ​​x 0 and y 0 into the right side of the equation y"=f(x,y) and calculate the slope y"=f(x 0,y 0) of the tangent to the integral curve at the point (x 0;y 0). To find the approximate value y 1 of the desired solution, we replace the integral curve on the segment [x 0 , x 1 ,] with a segment of its tangent at the point (x 0 ; y 0). In this case we get

y 1 - y 0 =f(x 0 ;y 0)(x 1 - x 0),

from where, since x 0, x 1, y 0 are known, we find

y1 = y0+f(x0;y0)(x1 - x0).

Substituting the values ​​x 1 and y 1 into the right side of the equation y"=f(x,y), we calculate the slope y"=f(x 1,y 1) of the tangent to the integral curve at the point (x 1;y 1). Next, replacing the integral curve on the segment with a tangent segment, we find the approximate value of the solution y 2 at point x 2:

y 2 = y 1 +f(x 1 ;y 1)(x 2 - x 1)

In this equality, x 1, y 1, x 2 are known, and y 2 is expressed through them.

Similarly we find

y 3 = y 2 +f(x 2 ;y 2) ?x, …, y n = y n-1 +f(x n-1 ;y n-1) ?x

Thus, the desired integral curve in the form of a broken line was approximately constructed and approximate values ​​y i of the desired solution at points x i were obtained. In this case, the values ​​of i are calculated using the formula

y i = y i-1 +f(x i-1 ;y i-1) ?x (i=1,2, …, n).

The formula is the main calculation formula of the Euler method. Its accuracy is higher, the smaller the difference?x.

Euler's method refers to numerical methods that provide a solution in the form of a table of approximate values ​​of the desired function y(x). It is relatively rough and is used mainly for approximate calculations. However, the ideas underlying Euler's method are the starting point for a number of other methods.

The degree of accuracy of Euler's method is, generally speaking, low. There are much more accurate methods for approximate solving differential equations.