Date of Award
5-1936
Document Type
Thesis
Degree Name
Master of Science (MS)
First Advisor
R.G. Smith
Keywords
Differential equations, Linear
Recommended Citation
Courtright, Henrietta, "Differential operator D" (1936). Electronic Theses & Dissertations. 712.
https://digitalcommons.pittstate.edu/etd/712
COinS
Comments
The general solution of the linear differential equation with constant coefficients is obtained usually by assuming y=Ae^mx as a trial solution of the associated homogeneous equation. This assumed trial solution leads to n linearly independent solutions, where n is the order of the differential equation. Any linear combination of these n linearly independent functions forms a general solution of the associated homogenous equation called the complementary function. This is combined with any particular solution of the non-homogeneous equation to form the complete solution.
Any existence proof of the solution is so difficult that it is omitted in all elementary texts on differential equations. This results in a unfavorable reaction on the part of the inquisitive student. Why assume such a solution? Is the complete solution as general as possible?
Further, certain exceptional cases such as multiple roots of the auxiliary equation are solved by involved indirect methods. The operator D is used ordinarily only in certain short methods for finding the particular integral.
This thesis presenta a more complete study of the properties of the operators D and D+a than is customarily given. This inverse of operate D+a leads to a direct solution of the linear differential equation consisting of a complementary function and a particular integral. The existence is based on the solution of a linear equation of the first order. In case of multiple roots, the solution is obtained directly and with little difficulty.
The various forms for expressing the complementary function are discussed, and the use of operator D in several short methods for finding the particular integral is shown.