Note on Finding the Complementary Function of a Linear Differential Equation with Constant Coefficients when the Auxiliary Equation has Equal Roots

1906 ◽  
Vol 13 (2) ◽  
pp. 33
Author(s):  
B. F. Finkel
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Auxiliary Equation ◽  
Complementary Function ◽  
Constant Coefficients ◽  
Linear Differential

Linear differential equations with constant coefficients

2019 ◽  
Vol 103 (557) ◽  
pp. 257-264
Author(s):  
Bethany Fralick ◽  
Reginald Koo
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Auxiliary Equation ◽  
Real Solutions ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Complex Solutions ◽  
Image Position ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

We consider the second order homogeneous linear differential equation (H) $${ ay'' + by' + cy = 0 }$$ with real coefficients a, b, c, and a ≠ 0. The function y = emx is a solution if, and only if, m satisfies the auxiliary equation am2 + bm + c = 0. When the roots of this are the complex conjugates m = p ± iq, then y = e(p ± iq)x are complex solutions of (H). Nevertheless, real solutions are given by y = c1epx cos qx + c2epx sin qx.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

scholarly journals ELEMENTARY METHOD OF SUBSTANTIATION AND IDEOLOGICAL MEANING OF THE THEORY OF THE SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS AND TRIGONOMETRIC FREE TERMS

2021 ◽  
pp. 105-114
Author(s):  
Zh. A. Sartabanov ◽  
A. Kh. Zhumagaziyev ◽  
A. A. Duyussova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Teaching Method ◽  
Second Order ◽  
School Mathematics ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Constant Coefficients ◽  
Linear Differential

In the article, adapted to the school course, the second order linear differential equations with constant coefficients and trigonometric free terms are investigated. The basic elementary methodological approaches to solving the equation are given. The solutions of the second order linear differential equation with constant coefficients and trigonometric free terms are investigated, which is a model of many phenomena. In addition, the applied values of the equation and its solutions were noted. The results obtained are presented in the form of theorems. The main novelty of the study is that these results are proved and generalized by elementary methods. These conclusions are proved in the framework of the methods of high school mathematics. This theory, known in general mathematics, is fully adapted to the implementation in secondary school mathematics and developed with the help of new elementary techniques that are understandable to the student. The main purpose of the research is to develop methods for solving a non-uniform linear differential equation of the second order with a constant coefficient at a level that a schoolboy can master. The result will be the creation of a special course program on the basics of ordinary differential equations in secondary schools of the natural-mathematical direction, the preparation of appropriate content material and providing them with a simple teaching method.

scholarly journals Fourier Transformation and Stability of a Differential Equation on L 1 ℝ

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Hamid Rezaei ◽  
Zahra Zafarasa ◽  
Lotfollah Karimi
Keyword(s):  
Differential Equation ◽  
Fourier Transform ◽  
Linear Differential Equation ◽  
Fourier Transformation ◽  
Ulam Stability ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Fourier Transform

In the present paper, by the Fourier transform, we show that every linear differential equation with constant coefficients of n -th order has a solution in L 1 ℝ which is infinitely differentiable in ℝ ∖ 0 . Moreover the Hyers–Ulam stability of such equations on L 1 ℝ is investigated.

Sign in / Sign up

Export Citation Format

Share Document