scholarly journals Determining a Particular Solution for Systems of Linear Differential Equations with Constant Coefficients that Have on the Column of Free Terms Functions Such As ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx

2020 ◽  
Vol 26 (3) ◽  
pp. 55-64
Author(s):  
Vasile Căruţaşu ◽  
Ioan Virca
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
The Other ◽  
Other Hand ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential ◽  
The Way

AbstractThe determination of a particular solution for the systems of linear differential equations with constant coefficients that have on the column of free terms functions such as ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx, or/and eb·x·Q(x) sinnx, n ∈ N, is based on the expansion of cosn x and sinn x, and, on the other hand, on how a particular solution for a column of free terms with functions such as ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx, n ∈ N, looks like. We can also write the way a particular solution looks like when we have a combination of two or more functions on the form ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx, n ∈ N in the column of free terms.

scholarly journals Determining a Particular Solution for N-th Order Linear Differential Equations with Constant Coefficients that Have the Free Term in the Following Form ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx

2020 ◽  
Vol 26 (3) ◽  
pp. 49-54
Author(s):  
Vasile Căruţaşu ◽  
Alexandru Hampu
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
The Other ◽  
Free Term ◽  
Order Linear ◽  
Other Hand ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential ◽  
The Way

AbstractA particular solution for the n-th order linear differential equations with constant coefficients that are free of term such as P(x)· ea·x ·cosnx or/and Q(x)·eb·x· sinnx, n∈ N, can be determined based on two elements: the way in which cosnx and sinnx can develop, and, on the other hand on the way a particular solution for the free terms P(x) ·ea·x·cosnx or/and Q(x)·eb·x·sinmx, n ∈N is sought. We can, of course, write the way a particular solution looks also in the case we have a combination of the two terms or more terms of this kind.

scholarly journals Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients

2017 ◽  
Vol 23 (3) ◽  
pp. 24-29
Author(s):  
Vasile Căruțașu
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Free Term ◽  
Order Linear ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential

Abstract For n-th order linear differential equations with constant coefficients, the problem to be solved is related to determining a particular solution, and then, with the general solution of n-th homogeneous linear differential equation with constant coefficients attached, to write the general solution of n-th linear differential equation with the given constant coefficients. In all the works that deal with this issue three situations are analyzed: the situation in which the free term is a polynomial P(x), the situation in which the free term is like P(x)· eα·x and lastly, the situation in which the free term is like eω·x · (P(x)· cos(β·x)+ Q(x)·sin(β·x)). In this study we aim to analyze if the free term is a combination of the three cases mentioned.

scholarly journals Determining a Particular Solution for the Systems of Linear Differential Equations with Constant Coefficients

2017 ◽  
Vol 23 (3) ◽  
pp. 30-36
Author(s):  
Vasile Căruțașu
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Homogeneous System ◽  
Linear Differential Equations ◽  
Order Linear ◽  
System A ◽  
Variation Of Constants ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential

Abstract As with the n-th order linear differential equations with constant coefficients, the problem to be solved is related to determining a particular solution, and then, using the general solution of the attached homogeneous system of linear differential equations with constant coefficients, to write the general solution of the initially given system. For homogeneous systems of linear differential equations with constant coefficients, the determination of the general solution is the method of eliminating or reducing which make the system a linear differential equation of the same order as that of the system, and its methods of solving it applies or the method of own values and vectors. If the system is non-homogeneous, then we also have to determine a particular solution that can be done in the same way as in the case of n-th order differential equations with constant coefficients, if the method of reduction or elimination was used, or the method of variation of constants, regardless of the method used to determine the general solution of the attached homogenous system of linear differential equations with constant coefficients. Whichever method is used, determining a particular solution for a system of linear differential equations with constant coefficients is difficult, in this study being proposed a method similar to that of n-th order linear differential equations with constant coefficients.

Kvalitet pitanja u TV školi u nastavi prirode i društva

2021 ◽  
Author(s):  
Jelena Lukić ◽  
Keyword(s):  
Social Science ◽  
The Other ◽  
Educational Content ◽  
Educational Levels ◽  
School Year ◽  
Other Hand ◽  
Cognitive Levels ◽  
The Way

The aim of this paperis to determine the quality of TV classes World around us and Nature and Social Science,which were broadcasted on Radio Television of Serbia during the pandemic in the school year 2019/20. Although the work was indirect, so the immediate interaction between the teacher and student is missed. Therefore, teacher's questions were the way of establishing some kind of interaction in such classes organized in this manner. For this reason, we wanted to establish the types of questions that teachers were asking to students through small screens. Considering that the achievements of learning are based on Bloom's Taxonomy, we were analyzing sixteen TV classes and classified the questions the teachers asked according to cognitive area, on six educational levels. The results indicate that the most common were question within lower cognitive levels were (knowledge, understanding and application), and that there are no statistically significant differences in cognitive levels on questions asked between lower (1st and 2nd grade) and higher grades (3rd and 4th grade), on the other hand, on the classes of determination of educational content teachers were asking statistically significant quality questions compared to the classes of interpretation.

scholarly journals Space-time caustics

1986 ◽  
Vol 9 (3) ◽  
pp. 531-540 ◽  
Author(s):  
Arthur D. Gorman
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Series Solution ◽  
Asymptotic Series ◽  
Inhomogeneous Media ◽  
Linear Differential Equations ◽  
Spatially Inhomogeneous ◽  
Asymptotic Series Solution ◽  
Linear Differential

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of linear differential equations modelling wave propagation in spatially inhomogeneous media at caustic (turning) points. Here the formalism is adapted to determine a class of asymptotic solutions at caustic points for those equations modelling wave propagation in media with both spatial and temporal inhomogeneities. The analogous Schrodinger equation is also considered.

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