On The Transformation Of The Generalized Euler\u27s Differential Equation Of Order-N

This thesis deals with the transformation and solution of the generalized Euler Differential Equation of Order - n. This problem was suggested and supervised by Dr. A. D. Stewart, Head of the Department of Mathematics, Prairie View A & M College, Prairie View, Texas. The writer is greatly indebted to Dr. Stewart, who is an authority on differential equations, for his directions and suggestions during the preparation of this thesis. The problem is first to take an nth-order differential equation with linear coefficients and transform it into an nth-order differential equation with constant coefficients. Secondly, to find a solution to the differential equation with constant coefficients and finally to transform it back to the solution of the original equation. In Chapter two, we shall show solutions to two Euler\u27s differential equations of order-two. The first differential equation will have a linear coefficient of one term and the second differential equation will have a linear expression consisting of two terms as a coefficient. Associated with these two differential equations which are non-homogeneous in nature will be two homogeneous differential equations and their solutions. Chapter three and four will deal with solutions of an nth-order Euler Differential Equation having the same properties as the second-order differential equation. Associated with these two nth-order Euler Differential Equations will be corresponding solution of homogeneous type nth order differential equations

A study of attitudes toward and achievement in mathematics of prospective elementary school teachers under a classroom diagnostic instructional model

This study sought information regarding attitudes toward and achievement in mathematics of prospective elementary school teachers under a classroom diagnostic instructional model. The classroom diagnostic instructional model is a cyclical model for classroom instruction that includes seven parts as diagrammed below. The subjects treated under this classroom diagnostic instructional model were the forty-eight prospective elementary school teachers enrolled in the first course of mathematics for prospective elementary school teachers (Math 263) at Prairie View A&M University, Prairie View, Texas. These forty-eight students were (i) match-paired utilizing each subject's quantitative aptitude score from their American College Testing Examination (ACT) and (ii) randomly assigned one subject of each pair to experimental group E1 and the remaining subject of each pair to experimental group E2. The following questions were investigated: 1. Does the use of a classroom diagnostic instructional model employing three levels of abstraction affect attitudes and achievement of high and low quantitative aptitude groups of prospective elementary school teachers. 2. Does the use of a classroom diagnostic instructional model employing two levels of abstraction (semi-concrete/abstract) affect attitudes and achievement of high and low quantitative aptitude groups of prospective elementary school teachers. 3. How does the attitude toward and achievement in mathematics of prospective elementary school teachers exposed to a classroom diagnostic instructional model utilizing three levels of abstraction (concrete/semi-concrete/abstract) compare to prospective elementary school teachers exposed to the same classroom diagnostic instructional model utilizing two levels of abstraction (semi-concrete/abstract). 4. How statistically significant is the change in attitudes toward and achievement in mathematics of prospective elementary school teachers as a result of exposure to a classroom diagnostic instructional model. The concept studied was the concept of fractions. The operations studied were addition, subtraction, multiplication and division of fractions. The attitudinal components tested were affective, cognitive and action components. From analysis of the hypotheses tested, it may be concluded that: 1. Achievement as measured by the achievement test is significantly and positively affected by the use if a classroom doagnostic instructional model utilizing three levels of abstraction as compared to the same instructional model using two levels of abstraction (semi-concrete/abstract). 2. Attitudes as measured by the 'Attitudes Toward Mathematics' scale developed by Suydam and Trueblood are not affected by the use of a classroom diagnostic instructional model utilizing three levels of abstraction as compared to the same instructional model using two levels of abstraction. 3. The utilization of a classroom diagnostic instructional model containing three levels of abstraction is a viable model for teaching prospective elementary school teachers mathematics.Education, College o