Logarithmic Integration Method for Solving Some Classes of Differential Equations | Статья в журнале «Молодой ученый»

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Автор:

Рубрика: Математика

Опубликовано в Молодой учёный №16 (254) апрель 2019 г.

Дата публикации: 19.04.2019

Статья просмотрена: 60 раз

Библиографическое описание:

Пономаренко А. Н. Logarithmic Integration Method for Solving Some Classes of Differential Equations // Молодой ученый. — 2019. — №16. — С. 1-9. — URL https://moluch.ru/archive/254/58104/ (дата обращения: 10.12.2019).



In this article presents logarithmic methods for solving first order and second order differential equations.

Let , be Riemann integrable functions; , , , ; , , , … , , , - is an integration constant. The symbol between two formulas will mean that the second formula follows from the first one.

1. First order differential equations

1.1. Linear inhomogeneous first order differential equation [1], [2], [3]:

. (1.1)

Logarithmic integration method. In equation (1.1) the function is not identically zero. Then be not identically zero. Then with equation (1.1), consequently we get

,

,

, (1.2)

,

,

,

,

,

,

. (1.3)

Remark 1.1. A similar method can be used to obtain a solution of the equation (1.1) in the Cauchy form [3]:

, (1.4)

where is a given constant.

Indeed, the equation (1.2) is equivalent to the equation

, (1.5)

where is an integration constant. Let , where . Then the equation (1.5) can be represented as

,

,

,

,

,

,

,

. (1.6)

If in the equation (1.6) we let , then we have the formula (1.4).

1.2. Bernoulli differential equation [3]:

, (1.7)

where .

Logarithmic integration method. Let be not identically zero. Then from the equations (1.7) we obtain

,

,

,

,

,

, (1.8)

,

,

,

,

,

,

,

, (1.9)

,

,

. (1.10)

Remark 1.2. At the beginning of the course of the method, we assumed that be not identically zero. It follows that the equation (1.7) has a particular solution , if .

Remark 1.2.1.(The second version of the logarithmic method.) In the equations (1.7) we obtain

,

,

,

,

,

,

,

. (1.11)

The equation (1.11) is a linear inhomogeneous first order differential equation, with respect to the function . Its solution by the with formula (1.3), has the form

. (1.12)

The formula (1.12) implies the solution (1.10).

Remark 1.2.2.(The third version of the logarithmic method.) In the equations (1.8) we obtain

,

,

,

,

. (1.13)

The equation (1.13) is similar to the equation (1.9).

1.3. The equation of the form:

, (1.14)

where .

Logarithmic integration method. From the equation (1.14), we obtain

,

,

,

,

,

,

,

. (1.15)

The equation (1.15) is a linear inhomogeneous first order differential equation, with respect to the function . Its solution, by the formula (1.3), has the form

. (1.16)

Solving the equation (1.16), with respect to , we have

,

. (1.17)

Second order differential equation

2.1. Linear homogeneous second order differential equation [1], [3]:

, (2.1)

where , are real numbers.

Let be not identically zero. Then from the equations (2.1) we obtain

,

, (2.2)

because , ,

.

Let in the equation (2.2):

.

Then we have equation (2.2) in the form

, (2.3)

. (2.4)

Case 1. . In this case we have equation (2.4) has be form

,

,

. (2.5)

Let in the equation (2.5): , . Then we obtain

,

, ,

,

, ,

. (2.6)

Returning to the change of variables , , in the equation (2.6), we obtain

,

,

, ,

. (2.7)

Since , then we have in the equation (2.7)

,

,

,

, (2.8)

where , is an integration constant.

Case 2. . In this case we have equation (2.4) has be form

.

Step by step from the last equation we obtain

,

, ,

. (2.9)

Since , then we have in the equation (2.9)

,

,

, (2.10)

where is an integration constant.

Case 3. . In this case we have equation (2.4) has be form

,

,

,

, ,

,

. (2.11)

Since , then we have in the equation (2.11)

,

,

, (2.12)

where , is an integration constant.

The formulas (2.8), (2.10), (2.12) solve the equation (2.1) in the respective cases 1,2,3. This method makes it possible to obtain these solutions without applying a complex analysis and finding a solution in the form .

References:

  1. C. H. Edwards, D. E. Penny. Differential Equations and Boundary Value Problems: Computing and Modeling (Third Edition), (2010) — 708 p.

2. C. H. Edwards, D. E. Penny, D. Calvis. Elementary differential equations, — 632 p.

3. N. M. Matveev. Metodu integrirovaniya obiknovennih differentsialnih uravneniy, Izdatyelstvo leningradskoho universiteta, (1955) — 655 p.



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