Analytical Proof of the Solution to Second Order Linear Homogeneous Differential Equation

There is no gainsaying that the field of differential equations opened up in numerable fields of research which hitherto remains yet to be fully explored in spite of the remarkable achievements made over the centuries. In this study, the subject is given renewed interest with a view to addressing a question that is considered germane to the field of differential equations (DEs). Over the centuries, the field of ordinary differential equations (ODEs) has received varied interest from mathematicians across the globe. One problem that has remained unresolved hitherto is a deductive proof of the solution to linear homogeneous differential equations of order 2, 3 or more. Solutions to this set of equations usually come in form of intuitive assumptions such as let y = e^rx or y = x^r, which is later confirmed by direct substitution. This study aims to provide a deductive solution leading to the proof that y = e^rx and y = x^r are not just intuitive assumptions but are indeed accurate in every sense of the word.